On Games Inspired by Gambler's Ruin with Unlimited Credit with many Gambling Scenarios By Shalosh B. Ekhad In this book we will analyze ALL possible games given by sets of steps each \ of them between 1 and , 4, in absolute value and each containing at least one positive and at least one negative step, an\ d once the set is decided every step is equally likely. We only treat the case where the sum of the s\ teps is not 0 For the set, {-1, 2} We have the following propositions --------------------------------------------- Proposition Number, 1 Consider a casino with a strange roulette, where there were, 2, possible outcomes as follows. Regarding losing You lose, 1, dollars with probability, 1/2 Regarding winning You win, 2, dollars with probability, 1/2 You start with a capital of 0 dollars, and have unlimited credit. Your goal \ in life is to gain a POSITIVE amount, and you quit as soon as you reach that goal. Note that the expected gain in ONE round is, 1/2 Since it is positive, sooner or later you will reach your goal. The probability that the goal is reached within, 300, rounds is , 0.99999999992478859120 The approximate expectation, variance, and scaled moments up to the, 4, -th are [3.2360679539390012400, 25.969186870570338784, 5.3036669202832661545, 49.123011672342980154] Let's compare it to the results of simulating it, 500, times. Of course this changes from time to time and the fraction of the simulated games that for which the goal was reached \ within, 300, rounds is , 1. The crude approximations for the expectation, variance etc. are [3.1320000000000000000, 22.126576000000000000, 4.3165968445850782855, 27.680886617510152485] So far we considered the Solitaire game. Now suppose that two players take t\ urns, and the first player to reach the goal of having positive capital is declared the winner. Assuming that th\ ey each are give at most, 300, turns the probability of the first-to-move player winning is 0.66108609130528843276 This ends Proposition No., 1, that took, 0.169, seconds to generate. --------------------------------------------- For the set, {-1, 3} We have the following propositions --------------------------------------------- Proposition Number, 2 Consider a casino with a strange roulette, where there were, 2, possible outcomes as follows. Regarding losing You lose, 1, dollars with probability, 1/2 Regarding winning You win, 3, dollars with probability, 1/2 You start with a capital of 0 dollars, and have unlimited credit. Your goal \ in life is to gain a POSITIVE amount, and you quit as soon as you reach that goal. Note that the expected gain in ONE round is, 1 Since it is positive, sooner or later you will reach your goal. The probability that the goal is reached within, 300, rounds is , 0.99999999999999999999 The approximate expectation, variance, and scaled moments up to the, 4, -th are [2.3829757679062374913, 7.3303351464613770388, 4.4212921593686975790, 34.573100877086844959] Let's compare it to the results of simulating it, 500, times. Of course this changes from time to time and the fraction of the simulated games that for which the goal was reached \ within, 300, rounds is , 1. The crude approximations for the expectation, variance etc. are [2.4920000000000000000, 9.5179360000000000000, 5.2343277768391785301, 41.495676974563391547] So far we considered the Solitaire game. Now suppose that two players take t\ urns, and the first player to reach the goal of having positive capital is declared the winner. Assuming that th\ ey each are give at most, 300, turns the probability of the first-to-move player winning is 0.66542479602204105091 This ends Proposition No., 2, that took, 0.119, seconds to generate. --------------------------------------------- For the set, {-1, 4} We have the following propositions --------------------------------------------- Proposition Number, 3 Consider a casino with a strange roulette, where there were, 2, possible outcomes as follows. Regarding losing You lose, 1, dollars with probability, 1/2 Regarding winning You win, 4, dollars with probability, 1/2 You start with a capital of 0 dollars, and have unlimited credit. Your goal \ in life is to gain a POSITIVE amount, and you quit as soon as you reach that goal. Note that the expected gain in ONE round is, 3/2 Since it is positive, sooner or later you will reach your goal. The probability that the goal is reached within, 300, rounds is , 1.0000000000000000000 The approximate expectation, variance, and scaled moments up to the, 4, -th are [2.1561901553356811691, 4.1379708092392878498, 3.9889999208343582652, 29.026175183252268991] Let's compare it to the results of simulating it, 500, times. Of course this changes from time to time and the fraction of the simulated games that for which the goal was reached \ within, 300, rounds is , 1. The crude approximations for the expectation, variance etc. are [2.3220000000000000000, 6.4183160000000000000, 4.3927856458083683494, 30.730069360617000175] So far we considered the Solitaire game. Now suppose that two players take t\ urns, and the first player to reach the goal of having positive capital is declared the winner. Assuming that th\ ey each are give at most, 300, turns the probability of the first-to-move player winning is 0.66636978642966486131 This ends Proposition No., 3, that took, 0.118, seconds to generate. --------------------------------------------- For the set, {-1, 1, 2} We have the following propositions --------------------------------------------- Proposition Number, 4 Consider a casino with a strange roulette, where there were, 3, possible outcomes as follows. Regarding losing You lose, 1, dollars with probability, 1/3 Regarding winning You win, 1, dollars with probability, 1/3 You win, 2, dollars with probability, 1/3 You start with a capital of 0 dollars, and have unlimited credit. Your goal \ in life is to gain a POSITIVE amount, and you quit as soon as you reach that goal. Note that the expected gain in ONE round is, 2/3 Since it is positive, sooner or later you will reach your goal. The probability that the goal is reached within, 300, rounds is , 1.0000000000000000000 The approximate expectation, variance, and scaled moments up to the, 4, -th are [2.1213203435596425730, 6.4926406871192850823, 4.4402742682694949540, 34.627624632935789768] Let's compare it to the results of simulating it, 500, times. Of course this changes from time to time and the fraction of the simulated games that for which the goal was reached \ within, 300, rounds is , 1. The crude approximations for the expectation, variance etc. are [2.1620000000000000000, 6.4157560000000000000, 3.4102980935956793165, 16.496582703438289482] So far we considered the Solitaire game. Now suppose that two players take t\ urns, and the first player to reach the goal of having positive capital is declared the winner. Assuming that th\ ey each are give at most, 300, turns the probability of the first-to-move player winning is 0.73306139256853332967 This ends Proposition No., 4, that took, 0.665, seconds to generate. --------------------------------------------- For the set, {-1, 1, 3} We have the following propositions --------------------------------------------- Proposition Number, 5 Consider a casino with a strange roulette, where there were, 3, possible outcomes as follows. Regarding losing You lose, 1, dollars with probability, 1/3 Regarding winning You win, 1, dollars with probability, 1/3 You win, 3, dollars with probability, 1/3 You start with a capital of 0 dollars, and have unlimited credit. Your goal \ in life is to gain a POSITIVE amount, and you quit as soon as you reach that goal. Note that the expected gain in ONE round is, 1 Since it is positive, sooner or later you will reach your goal. The probability that the goal is reached within, 300, rounds is , 1.0000000000000000000 The approximate expectation, variance, and scaled moments up to the, 4, -th are [1.9394650585867228909, 4.0121621637987979564, 3.9954452052224542360, 28.525117884237564259] Let's compare it to the results of simulating it, 500, times. Of course this changes from time to time and the fraction of the simulated games that for which the goal was reached \ within, 300, rounds is , 1. The crude approximations for the expectation, variance etc. are [1.8380000000000000000, 3.1397560000000000000, 3.6754512015637521889, 22.523007007846212594] So far we considered the Solitaire game. Now suppose that two players take t\ urns, and the first player to reach the goal of having positive capital is declared the winner. Assuming that th\ ey each are give at most, 300, turns the probability of the first-to-move player winning is 0.73577397190760204784 This ends Proposition No., 5, that took, 0.367, seconds to generate. --------------------------------------------- For the set, {-1, 1, 4} We have the following propositions --------------------------------------------- Proposition Number, 6 Consider a casino with a strange roulette, where there were, 3, possible outcomes as follows. Regarding losing You lose, 1, dollars with probability, 1/3 Regarding winning You win, 1, dollars with probability, 1/3 You win, 4, dollars with probability, 1/3 You start with a capital of 0 dollars, and have unlimited credit. Your goal \ in life is to gain a POSITIVE amount, and you quit as soon as you reach that goal. Note that the expected gain in ONE round is, 4/3 Since it is positive, sooner or later you will reach your goal. The probability that the goal is reached within, 300, rounds is , 1.0000000000000000000 The approximate expectation, variance, and scaled moments up to the, 4, -th are [1.8843596232155118823, 3.2573889932959658914, 3.6476565904816650126, 24.283487216331886497] Let's compare it to the results of simulating it, 500, times. Of course this changes from time to time and the fraction of the simulated games that for which the goal was reached \ within, 300, rounds is , 1. The crude approximations for the expectation, variance etc. are [1.9580000000000000000, 3.3722360000000000000, 2.8309301274110773306, 12.867026250930623342] So far we considered the Solitaire game. Now suppose that two players take t\ urns, and the first player to reach the goal of having positive capital is declared the winner. Assuming that th\ ey each are give at most, 300, turns the probability of the first-to-move player winning is 0.73586833085816195311 This ends Proposition No., 6, that took, 0.652, seconds to generate. --------------------------------------------- For the set, {-1, 2, 3} We have the following propositions --------------------------------------------- Proposition Number, 7 Consider a casino with a strange roulette, where there were, 3, possible outcomes as follows. Regarding losing You lose, 1, dollars with probability, 1/3 Regarding winning You win, 2, dollars with probability, 1/3 You win, 3, dollars with probability, 1/3 You start with a capital of 0 dollars, and have unlimited credit. Your goal \ in life is to gain a POSITIVE amount, and you quit as soon as you reach that goal. Note that the expected gain in ONE round is, 4/3 Since it is positive, sooner or later you will reach your goal. The probability that the goal is reached within, 300, rounds is , 1.0000000000000000000 The approximate expectation, variance, and scaled moments up to the, 4, -th are [1.6382903785557460002, 1.8822226405987295457, 4.0921549648234134256, 29.053911670931404624] Let's compare it to the results of simulating it, 500, times. Of course this changes from time to time and the fraction of the simulated games that for which the goal was reached \ within, 300, rounds is , 1. The crude approximations for the expectation, variance etc. are [1.5820000000000000000, 1.6152760000000000000, 4.2732446581848397099, 28.983710272532135583] So far we considered the Solitaire game. Now suppose that two players take t\ urns, and the first player to reach the goal of having positive capital is declared the winner. Assuming that th\ ey each are give at most, 300, turns the probability of the first-to-move player winning is 0.74827779942779937753 This ends Proposition No., 7, that took, 0.673, seconds to generate. --------------------------------------------- For the set, {-1, 2, 4} We have the following propositions --------------------------------------------- Proposition Number, 8 Consider a casino with a strange roulette, where there were, 3, possible outcomes as follows. Regarding losing You lose, 1, dollars with probability, 1/3 Regarding winning You win, 2, dollars with probability, 1/3 You win, 4, dollars with probability, 1/3 You start with a capital of 0 dollars, and have unlimited credit. Your goal \ in life is to gain a POSITIVE amount, and you quit as soon as you reach that goal. Note that the expected gain in ONE round is, 5/3 Since it is positive, sooner or later you will reach your goal. The probability that the goal is reached within, 300, rounds is , 1.0000000000000000000 The approximate expectation, variance, and scaled moments up to the, 4, -th are [1.6101929566028330836, 1.5973772823646937717, 3.8441612872771645084, 26.160981497065307942] Let's compare it to the results of simulating it, 500, times. Of course this changes from time to time and the fraction of the simulated games that for which the goal was reached \ within, 300, rounds is , 1. The crude approximations for the expectation, variance etc. are [1.5060000000000000000, 1.0339640000000000000, 2.8941957549454250823, 12.868286144960717792] So far we considered the Solitaire game. Now suppose that two players take t\ urns, and the first player to reach the goal of having positive capital is declared the winner. Assuming that th\ ey each are give at most, 300, turns the probability of the first-to-move player winning is 0.74847068594521860107 This ends Proposition No., 8, that took, 0.614, seconds to generate. --------------------------------------------- For the set, {-1, 3, 4} We have the following propositions --------------------------------------------- Proposition Number, 9 Consider a casino with a strange roulette, where there were, 3, possible outcomes as follows. Regarding losing You lose, 1, dollars with probability, 1/3 Regarding winning You win, 3, dollars with probability, 1/3 You win, 4, dollars with probability, 1/3 You start with a capital of 0 dollars, and have unlimited credit. Your goal \ in life is to gain a POSITIVE amount, and you quit as soon as you reach that goal. Note that the expected gain in ONE round is, 2 Since it is positive, sooner or later you will reach your goal. The probability that the goal is reached within, 300, rounds is , 1.0000000000000000000 The approximate expectation, variance, and scaled moments up to the, 4, -th are [1.5402691138989976488, 1.1227523412478583342, 3.7310139231627979003, 25.684909148191582839] Let's compare it to the results of simulating it, 500, times. Of course this changes from time to time and the fraction of the simulated games that for which the goal was reached \ within, 300, rounds is , 1. The crude approximations for the expectation, variance etc. are [1.4880000000000000000, 0.90985600000000000000, 3.9127090175681592397, 28.936472948430892054] So far we considered the Solitaire game. Now suppose that two players take t\ urns, and the first player to reach the goal of having positive capital is declared the winner. Assuming that th\ ey each are give at most, 300, turns the probability of the first-to-move player winning is 0.74981312995233764766 This ends Proposition No., 9, that took, 0.606, seconds to generate. --------------------------------------------- For the set, {-1, 1, 2, 3} We have the following propositions --------------------------------------------- Proposition Number, 10 Consider a casino with a strange roulette, where there were, 4, possible outcomes as follows. Regarding losing You lose, 1, dollars with probability, 1/4 Regarding winning You win, 1, dollars with probability, 1/4 You win, 2, dollars with probability, 1/4 You win, 3, dollars with probability, 1/4 You start with a capital of 0 dollars, and have unlimited credit. Your goal \ in life is to gain a POSITIVE amount, and you quit as soon as you reach that goal. Note that the expected gain in ONE round is, 5/4 Since it is positive, sooner or later you will reach your goal. The probability that the goal is reached within, 300, rounds is , 1.0000000000000000000 The approximate expectation, variance, and scaled moments up to the, 4, -th are [1.5224378306902545027, 1.5203038341262403311, 3.9450000057882531040, 26.806285787258183807] Let's compare it to the results of simulating it, 500, times. Of course this changes from time to time and the fraction of the simulated games that for which the goal was reached \ within, 300, rounds is , 1. The crude approximations for the expectation, variance etc. are [1.5340000000000000000, 1.6128440000000000000, 4.3734092497589722039, 32.139365270805029588] So far we considered the Solitaire game. Now suppose that two players take t\ urns, and the first player to reach the goal of having positive capital is declared the winner. Assuming that th\ ey each are give at most, 300, turns the probability of the first-to-move player winning is 0.79155908640598431013 This ends Proposition No., 10, that took, 0.591, seconds to generate. --------------------------------------------- For the set, {-1, 1, 2, 4} We have the following propositions --------------------------------------------- Proposition Number, 11 Consider a casino with a strange roulette, where there were, 4, possible outcomes as follows. Regarding losing You lose, 1, dollars with probability, 1/4 Regarding winning You win, 1, dollars with probability, 1/4 You win, 2, dollars with probability, 1/4 You win, 4, dollars with probability, 1/4 You start with a capital of 0 dollars, and have unlimited credit. Your goal \ in life is to gain a POSITIVE amount, and you quit as soon as you reach that goal. Note that the expected gain in ONE round is, 3/2 Since it is positive, sooner or later you will reach your goal. The probability that the goal is reached within, 300, rounds is , 1.0000000000000000000 The approximate expectation, variance, and scaled moments up to the, 4, -th are [1.5124702552128152486, 1.4156296207946855858, 3.7826251547766944362, 24.758841807383165479] Let's compare it to the results of simulating it, 500, times. Of course this changes from time to time and the fraction of the simulated games that for which the goal was reached \ within, 300, rounds is , 1. The crude approximations for the expectation, variance etc. are [1.5040000000000000000, 1.3659840000000000000, 4.1431981260370476979, 27.479606334098605164] So far we considered the Solitaire game. Now suppose that two players take t\ urns, and the first player to reach the goal of having positive capital is declared the winner. Assuming that th\ ey each are give at most, 300, turns the probability of the first-to-move player winning is 0.79163353749759070111 This ends Proposition No., 11, that took, 0.640, seconds to generate. --------------------------------------------- For the set, {-1, 1, 3, 4} We have the following propositions --------------------------------------------- Proposition Number, 12 Consider a casino with a strange roulette, where there were, 4, possible outcomes as follows. Regarding losing You lose, 1, dollars with probability, 1/4 Regarding winning You win, 1, dollars with probability, 1/4 You win, 3, dollars with probability, 1/4 You win, 4, dollars with probability, 1/4 You start with a capital of 0 dollars, and have unlimited credit. Your goal \ in life is to gain a POSITIVE amount, and you quit as soon as you reach that goal. Note that the expected gain in ONE round is, 7/4 Since it is positive, sooner or later you will reach your goal. The probability that the goal is reached within, 300, rounds is , 1.0000000000000000000 The approximate expectation, variance, and scaled moments up to the, 4, -th are [1.4786675209071412711, 1.1641070527482374824, 3.6002145214639289606, 22.912859664335110734] Let's compare it to the results of simulating it, 500, times. Of course this changes from time to time and the fraction of the simulated games that for which the goal was reached \ within, 300, rounds is , 1. The crude approximations for the expectation, variance etc. are [1.4520000000000000000, 1.0556960000000000000, 3.6472230994871895439, 21.013551381536465730] So far we considered the Solitaire game. Now suppose that two players take t\ urns, and the first player to reach the goal of having positive capital is declared the winner. Assuming that th\ ey each are give at most, 300, turns the probability of the first-to-move player winning is 0.79243430323253081016 This ends Proposition No., 12, that took, 0.588, seconds to generate. --------------------------------------------- For the set, {-1, 2, 3, 4} We have the following propositions --------------------------------------------- Proposition Number, 13 Consider a casino with a strange roulette, where there were, 4, possible outcomes as follows. Regarding losing You lose, 1, dollars with probability, 1/4 Regarding winning You win, 2, dollars with probability, 1/4 You win, 3, dollars with probability, 1/4 You win, 4, dollars with probability, 1/4 You start with a capital of 0 dollars, and have unlimited credit. Your goal \ in life is to gain a POSITIVE amount, and you quit as soon as you reach that goal. Note that the expected gain in ONE round is, 2 Since it is positive, sooner or later you will reach your goal. The probability that the goal is reached within, 300, rounds is , 1.0000000000000000000 The approximate expectation, variance, and scaled moments up to the, 4, -th are [1.3727905325136581932, 0.71224072940374084797, 3.8750872020715340356, 26.267509327945371169] Let's compare it to the results of simulating it, 500, times. Of course this changes from time to time and the fraction of the simulated games that for which the goal was reached \ within, 300, rounds is , 1. The crude approximations for the expectation, variance etc. are [1.4580000000000000000, 0.95623600000000000000, 2.9602462337262507475, 12.737939704149007887] So far we considered the Solitaire game. Now suppose that two players take t\ urns, and the first player to reach the goal of having positive capital is declared the winner. Assuming that th\ ey each are give at most, 300, turns the probability of the first-to-move player winning is 0.79948540473370319517 This ends Proposition No., 13, that took, 0.590, seconds to generate. --------------------------------------------- For the set, {-1, 1, 2, 3, 4} We have the following propositions --------------------------------------------- Proposition Number, 14 Consider a casino with a strange roulette, where there were, 5, possible outcomes as follows. Regarding losing You lose, 1, dollars with probability, 1/5 Regarding winning You win, 1, dollars with probability, 1/5 You win, 2, dollars with probability, 1/5 You win, 3, dollars with probability, 1/5 You win, 4, dollars with probability, 1/5 You start with a capital of 0 dollars, and have unlimited credit. Your goal \ in life is to gain a POSITIVE amount, and you quit as soon as you reach that goal. Note that the expected gain in ONE round is, 9/5 Since it is positive, sooner or later you will reach your goal. The probability that the goal is reached within, 300, rounds is , 1.0000000000000000000 The approximate expectation, variance, and scaled moments up to the, 4, -th are [1.3395394543761710887, 0.72984929534213589382, 3.8388737959201166516, 24.782110122032072341] Let's compare it to the results of simulating it, 500, times. Of course this changes from time to time and the fraction of the simulated games that for which the goal was reached \ within, 300, rounds is , 1. The crude approximations for the expectation, variance etc. are [1.3660000000000000000, 0.97604400000000000000, 5.1906759626017513654, 41.907680733270532704] So far we considered the Solitaire game. Now suppose that two players take t\ urns, and the first player to reach the goal of having positive capital is declared the winner. Assuming that th\ ey each are give at most, 300, turns the probability of the first-to-move player winning is 0.82854428484209252882 This ends Proposition No., 14, that took, 0.764, seconds to generate. --------------------------------------------- For the set, {-2, 3} We have the following propositions --------------------------------------------- Proposition Number, 15 Consider a casino with a strange roulette, where there were, 2, possible outcomes as follows. Regarding losing You lose, 2, dollars with probability, 1/2 Regarding winning You win, 3, dollars with probability, 1/2 You start with a capital of 0 dollars, and have unlimited credit. Your goal \ in life is to gain a POSITIVE amount, and you quit as soon as you reach that goal. Note that the expected gain in ONE round is, 1/2 Since it is positive, sooner or later you will reach your goal. The probability that the goal is reached within, 300, rounds is , 0.99999262868191959970 The approximate expectation, variance, and scaled moments up to the, 4, -th are [4.4507513685529168264, 100.89775936041966347, 7.2482338874721936634, 86.011684162674100935] Let's compare it to the results of simulating it, 500, times. Of course this changes from time to time and the fraction of the simulated games that for which the goal was reached \ within, 300, rounds is , 1. The crude approximations for the expectation, variance etc. are [5.1320000000000000000, 114.71057600000000000, 5.9135677312621535966, 54.480637851053328777] So far we considered the Solitaire game. Now suppose that two players take t\ urns, and the first player to reach the goal of having positive capital is declared the winner. Assuming that th\ ey each are give at most, 300, turns the probability of the first-to-move player winning is 0.65995123063516973530 This ends Proposition No., 15, that took, 0.202, seconds to generate. --------------------------------------------- For the set, {-2, 4} We have the following propositions --------------------------------------------- Proposition Number, 16 Consider a casino with a strange roulette, where there were, 2, possible outcomes as follows. Regarding losing You lose, 2, dollars with probability, 1/2 Regarding winning You win, 4, dollars with probability, 1/2 You start with a capital of 0 dollars, and have unlimited credit. Your goal \ in life is to gain a POSITIVE amount, and you quit as soon as you reach that goal. Note that the expected gain in ONE round is, 1 Since it is positive, sooner or later you will reach your goal. The probability that the goal is reached within, 300, rounds is , 0.99999999992478859120 The approximate expectation, variance, and scaled moments up to the, 4, -th are [3.2360679539390012400, 25.969186870570338784, 5.3036669202832661545, 49.123011672342980154] Let's compare it to the results of simulating it, 500, times. Of course this changes from time to time and the fraction of the simulated games that for which the goal was reached \ within, 300, rounds is , 1. The crude approximations for the expectation, variance etc. are [2.9900000000000000000, 16.005900000000000000, 3.8632094965196122434, 22.894376673050297324] So far we considered the Solitaire game. Now suppose that two players take t\ urns, and the first player to reach the goal of having positive capital is declared the winner. Assuming that th\ ey each are give at most, 300, turns the probability of the first-to-move player winning is 0.66108609130528843276 This ends Proposition No., 16, that took, 0.161, seconds to generate. --------------------------------------------- For the set, {-2, 1, 2} We have the following propositions --------------------------------------------- Proposition Number, 17 Consider a casino with a strange roulette, where there were, 3, possible outcomes as follows. Regarding losing You lose, 2, dollars with probability, 1/3 Regarding winning You win, 1, dollars with probability, 1/3 You win, 2, dollars with probability, 1/3 You start with a capital of 0 dollars, and have unlimited credit. Your goal \ in life is to gain a POSITIVE amount, and you quit as soon as you reach that goal. Note that the expected gain in ONE round is, 1/3 Since it is positive, sooner or later you will reach your goal. The probability that the goal is reached within, 300, rounds is , 0.99998893219203375184 The approximate expectation, variance, and scaled moments up to the, 4, -th are [4.3313585940495090208, 106.69659493243132313, 7.4854954879301997738, 90.774583912023952633] Let's compare it to the results of simulating it, 500, times. Of course this changes from time to time and the fraction of the simulated games that for which the goal was reached \ within, 300, rounds is , 1. The crude approximations for the expectation, variance etc. are [4.4340000000000000000, 122.39764400000000000, 7.7722650247274764479, 81.213533484913260628] So far we considered the Solitaire game. Now suppose that two players take t\ urns, and the first player to reach the goal of having positive capital is declared the winner. Assuming that th\ ey each are give at most, 300, turns the probability of the first-to-move player winning is 0.73000272688246834578 This ends Proposition No., 17, that took, 1.335, seconds to generate. --------------------------------------------- For the set, {-2, 1, 3} We have the following propositions --------------------------------------------- Proposition Number, 18 Consider a casino with a strange roulette, where there were, 3, possible outcomes as follows. Regarding losing You lose, 2, dollars with probability, 1/3 Regarding winning You win, 1, dollars with probability, 1/3 You win, 3, dollars with probability, 1/3 You start with a capital of 0 dollars, and have unlimited credit. Your goal \ in life is to gain a POSITIVE amount, and you quit as soon as you reach that goal. Note that the expected gain in ONE round is, 2/3 Since it is positive, sooner or later you will reach your goal. The probability that the goal is reached within, 300, rounds is , 0.99999999976781894926 The approximate expectation, variance, and scaled moments up to the, 4, -th are [2.7587353506017930129, 23.275709598887906617, 6.1023875898148718987, 64.005255654445095365] Let's compare it to the results of simulating it, 500, times. Of course this changes from time to time and the fraction of the simulated games that for which the goal was reached \ within, 300, rounds is , 1. The crude approximations for the expectation, variance etc. are [2.7700000000000000000, 19.017100000000000000, 4.1427733342080435169, 25.252248553047729055] So far we considered the Solitaire game. Now suppose that two players take t\ urns, and the first player to reach the goal of having positive capital is declared the winner. Assuming that th\ ey each are give at most, 300, turns the probability of the first-to-move player winning is 0.73096326405941944769 This ends Proposition No., 18, that took, 1.332, seconds to generate. --------------------------------------------- For the set, {-2, 1, 4} We have the following propositions --------------------------------------------- Proposition Number, 19 Consider a casino with a strange roulette, where there were, 3, possible outcomes as follows. Regarding losing You lose, 2, dollars with probability, 1/3 Regarding winning You win, 1, dollars with probability, 1/3 You win, 4, dollars with probability, 1/3 You start with a capital of 0 dollars, and have unlimited credit. Your goal \ in life is to gain a POSITIVE amount, and you quit as soon as you reach that goal. Note that the expected gain in ONE round is, 1 Since it is positive, sooner or later you will reach your goal. The probability that the goal is reached within, 300, rounds is , 0.99999999999999146887 The approximate expectation, variance, and scaled moments up to the, 4, -th are [2.3870792593400375173, 11.839613753972780606, 5.2465063559038866639, 47.931926102847450491] Let's compare it to the results of simulating it, 500, times. Of course this changes from time to time and the fraction of the simulated games that for which the goal was reached \ within, 300, rounds is , 1. The crude approximations for the expectation, variance etc. are [2.1380000000000000000, 8.2589560000000000000, 4.5079644211152813918, 27.731157172244316279] So far we considered the Solitaire game. Now suppose that two players take t\ urns, and the first player to reach the goal of having positive capital is declared the winner. Assuming that th\ ey each are give at most, 300, turns the probability of the first-to-move player winning is 0.73256313427492398127 This ends Proposition No., 19, that took, 0.531, seconds to generate. --------------------------------------------- For the set, {-2, 2, 3} We have the following propositions --------------------------------------------- Proposition Number, 20 Consider a casino with a strange roulette, where there were, 3, possible outcomes as follows. Regarding losing You lose, 2, dollars with probability, 1/3 Regarding winning You win, 2, dollars with probability, 1/3 You win, 3, dollars with probability, 1/3 You start with a capital of 0 dollars, and have unlimited credit. Your goal \ in life is to gain a POSITIVE amount, and you quit as soon as you reach that goal. Note that the expected gain in ONE round is, 1 Since it is positive, sooner or later you will reach your goal. The probability that the goal is reached within, 300, rounds is , 0.99999999999999994661 The approximate expectation, variance, and scaled moments up to the, 4, -th are [2.2078666375504980149, 8.6578805136381473768, 5.1681852129501965884, 46.577591045101511088] Let's compare it to the results of simulating it, 500, times. Of course this changes from time to time and the fraction of the simulated games that for which the goal was reached \ within, 300, rounds is , 1. The crude approximations for the expectation, variance etc. are [2.2760000000000000000, 10.139824000000000000, 5.6954070816295575816, 51.482277511445609829] So far we considered the Solitaire game. Now suppose that two players take t\ urns, and the first player to reach the goal of having positive capital is declared the winner. Assuming that th\ ey each are give at most, 300, turns the probability of the first-to-move player winning is 0.73301247397892596367 This ends Proposition No., 20, that took, 1.197, seconds to generate. --------------------------------------------- For the set, {-2, 2, 4} We have the following propositions --------------------------------------------- Proposition Number, 21 Consider a casino with a strange roulette, where there were, 3, possible outcomes as follows. Regarding losing You lose, 2, dollars with probability, 1/3 Regarding winning You win, 2, dollars with probability, 1/3 You win, 4, dollars with probability, 1/3 You start with a capital of 0 dollars, and have unlimited credit. Your goal \ in life is to gain a POSITIVE amount, and you quit as soon as you reach that goal. Note that the expected gain in ONE round is, 4/3 Since it is positive, sooner or later you will reach your goal. The probability that the goal is reached within, 300, rounds is , 1.0000000000000000000 The approximate expectation, variance, and scaled moments up to the, 4, -th are [2.1213203435596425730, 6.4926406871192850823, 4.4402742682694949540, 34.627624632935789768] Let's compare it to the results of simulating it, 500, times. Of course this changes from time to time and the fraction of the simulated games that for which the goal was reached \ within, 300, rounds is , 1. The crude approximations for the expectation, variance etc. are [1.9860000000000000000, 5.0338040000000000000, 4.3121831115668964005, 30.173481465701141855] So far we considered the Solitaire game. Now suppose that two players take t\ urns, and the first player to reach the goal of having positive capital is declared the winner. Assuming that th\ ey each are give at most, 300, turns the probability of the first-to-move player winning is 0.73306139256853332967 This ends Proposition No., 21, that took, 0.713, seconds to generate. --------------------------------------------- For the set, {-2, 3, 4} We have the following propositions --------------------------------------------- Proposition Number, 22 Consider a casino with a strange roulette, where there were, 3, possible outcomes as follows. Regarding losing You lose, 2, dollars with probability, 1/3 Regarding winning You win, 3, dollars with probability, 1/3 You win, 4, dollars with probability, 1/3 You start with a capital of 0 dollars, and have unlimited credit. Your goal \ in life is to gain a POSITIVE amount, and you quit as soon as you reach that goal. Note that the expected gain in ONE round is, 5/3 Since it is positive, sooner or later you will reach your goal. The probability that the goal is reached within, 300, rounds is , 1.0000000000000000000 The approximate expectation, variance, and scaled moments up to the, 4, -th are [1.7620027290950534818, 3.2208113034214304017, 4.7278461110444798712, 38.373116463038280067] Let's compare it to the results of simulating it, 500, times. Of course this changes from time to time and the fraction of the simulated games that for which the goal was reached \ within, 300, rounds is , 1. The crude approximations for the expectation, variance etc. are [1.8680000000000000000, 4.1105760000000000000, 4.9918451594250262140, 41.190190986182244870] So far we considered the Solitaire game. Now suppose that two players take t\ urns, and the first player to reach the goal of having positive capital is declared the winner. Assuming that th\ ey each are give at most, 300, turns the probability of the first-to-move player winning is 0.74856090782810838039 This ends Proposition No., 22, that took, 1.227, seconds to generate. --------------------------------------------- For the set, {-2, 1, 2, 3} We have the following propositions --------------------------------------------- Proposition Number, 23 Consider a casino with a strange roulette, where there were, 4, possible outcomes as follows. Regarding losing You lose, 2, dollars with probability, 1/4 Regarding winning You win, 1, dollars with probability, 1/4 You win, 2, dollars with probability, 1/4 You win, 3, dollars with probability, 1/4 You start with a capital of 0 dollars, and have unlimited credit. Your goal \ in life is to gain a POSITIVE amount, and you quit as soon as you reach that goal. Note that the expected gain in ONE round is, 1 Since it is positive, sooner or later you will reach your goal. The probability that the goal is reached within, 300, rounds is , 1.0000000000000000000 The approximate expectation, variance, and scaled moments up to the, 4, -th are [1.8835774996053437592, 5.4800869325198418900, 5.1018963084634341022, 44.683902695159420416] Let's compare it to the results of simulating it, 500, times. Of course this changes from time to time and the fraction of the simulated games that for which the goal was reached \ within, 300, rounds is , 1. The crude approximations for the expectation, variance etc. are [1.7300000000000000000, 3.3611000000000000000, 3.7682813333104775176, 20.878150848246814163] So far we considered the Solitaire game. Now suppose that two players take t\ urns, and the first player to reach the goal of having positive capital is declared the winner. Assuming that th\ ey each are give at most, 300, turns the probability of the first-to-move player winning is 0.78716717032649779075 This ends Proposition No., 23, that took, 1.110, seconds to generate. --------------------------------------------- For the set, {-2, 1, 2, 4} We have the following propositions --------------------------------------------- Proposition Number, 24 Consider a casino with a strange roulette, where there were, 4, possible outcomes as follows. Regarding losing You lose, 2, dollars with probability, 1/4 Regarding winning You win, 1, dollars with probability, 1/4 You win, 2, dollars with probability, 1/4 You win, 4, dollars with probability, 1/4 You start with a capital of 0 dollars, and have unlimited credit. Your goal \ in life is to gain a POSITIVE amount, and you quit as soon as you reach that goal. Note that the expected gain in ONE round is, 5/4 Since it is positive, sooner or later you will reach your goal. The probability that the goal is reached within, 300, rounds is , 1.0000000000000000000 The approximate expectation, variance, and scaled moments up to the, 4, -th are [1.8147253909298527277, 4.2603478801280663296, 4.6942058345021294678, 38.241935058768124653] Let's compare it to the results of simulating it, 500, times. Of course this changes from time to time and the fraction of the simulated games that for which the goal was reached \ within, 300, rounds is , 1. The crude approximations for the expectation, variance etc. are [2.0400000000000000000, 5.0424000000000000000, 3.0531918290511675853, 13.671478787063238247] So far we considered the Solitaire game. Now suppose that two players take t\ urns, and the first player to reach the goal of having positive capital is declared the winner. Assuming that th\ ey each are give at most, 300, turns the probability of the first-to-move player winning is 0.78741509012707064691 This ends Proposition No., 24, that took, 1.133, seconds to generate. --------------------------------------------- For the set, {-2, 1, 3, 4} We have the following propositions --------------------------------------------- Proposition Number, 25 Consider a casino with a strange roulette, where there were, 4, possible outcomes as follows. Regarding losing You lose, 2, dollars with probability, 1/4 Regarding winning You win, 1, dollars with probability, 1/4 You win, 3, dollars with probability, 1/4 You win, 4, dollars with probability, 1/4 You start with a capital of 0 dollars, and have unlimited credit. Your goal \ in life is to gain a POSITIVE amount, and you quit as soon as you reach that goal. Note that the expected gain in ONE round is, 3/2 Since it is positive, sooner or later you will reach your goal. The probability that the goal is reached within, 300, rounds is , 1.0000000000000000000 The approximate expectation, variance, and scaled moments up to the, 4, -th are [1.6453827303837755786, 2.8198735735972517010, 4.7541131071882358189, 38.309231602053046702] Let's compare it to the results of simulating it, 500, times. Of course this changes from time to time and the fraction of the simulated games that for which the goal was reached \ within, 300, rounds is , 1. The crude approximations for the expectation, variance etc. are [1.5700000000000000000, 2.5571000000000000000, 5.2491919011299356001, 43.149054725802378803] So far we considered the Solitaire game. Now suppose that two players take t\ urns, and the first player to reach the goal of having positive capital is declared the winner. Assuming that th\ ey each are give at most, 300, turns the probability of the first-to-move player winning is 0.79062119802776411048 This ends Proposition No., 25, that took, 1.107, seconds to generate. --------------------------------------------- For the set, {-2, 2, 3, 4} We have the following propositions --------------------------------------------- Proposition Number, 26 Consider a casino with a strange roulette, where there were, 4, possible outcomes as follows. Regarding losing You lose, 2, dollars with probability, 1/4 Regarding winning You win, 2, dollars with probability, 1/4 You win, 3, dollars with probability, 1/4 You win, 4, dollars with probability, 1/4 You start with a capital of 0 dollars, and have unlimited credit. Your goal \ in life is to gain a POSITIVE amount, and you quit as soon as you reach that goal. Note that the expected gain in ONE round is, 7/4 Since it is positive, sooner or later you will reach your goal. The probability that the goal is reached within, 300, rounds is , 1.0000000000000000000 The approximate expectation, variance, and scaled moments up to the, 4, -th are [1.5690605410868188545, 1.9727463981556947037, 4.3705390377930114188, 32.915707478166959650] Let's compare it to the results of simulating it, 500, times. Of course this changes from time to time and the fraction of the simulated games that for which the goal was reached \ within, 300, rounds is , 1. The crude approximations for the expectation, variance etc. are [1.5160000000000000000, 1.7137440000000000000, 3.9767698401471165507, 22.449678655607231936] So far we considered the Solitaire game. Now suppose that two players take t\ urns, and the first player to reach the goal of having positive capital is declared the winner. Assuming that th\ ey each are give at most, 300, turns the probability of the first-to-move player winning is 0.79108163855009393129 This ends Proposition No., 26, that took, 1.097, seconds to generate. --------------------------------------------- For the set, {-2, 1, 2, 3, 4} We have the following propositions --------------------------------------------- Proposition Number, 27 Consider a casino with a strange roulette, where there were, 5, possible outcomes as follows. Regarding losing You lose, 2, dollars with probability, 1/5 Regarding winning You win, 1, dollars with probability, 1/5 You win, 2, dollars with probability, 1/5 You win, 3, dollars with probability, 1/5 You win, 4, dollars with probability, 1/5 You start with a capital of 0 dollars, and have unlimited credit. Your goal \ in life is to gain a POSITIVE amount, and you quit as soon as you reach that goal. Note that the expected gain in ONE round is, 8/5 Since it is positive, sooner or later you will reach your goal. The probability that the goal is reached within, 300, rounds is , 1.0000000000000000000 The approximate expectation, variance, and scaled moments up to the, 4, -th are [1.4860854967033019513, 1.7532601123684382393, 4.5909677663319156628, 35.424156405008943306] Let's compare it to the results of simulating it, 500, times. Of course this changes from time to time and the fraction of the simulated games that for which the goal was reached \ within, 300, rounds is , 1. The crude approximations for the expectation, variance etc. are [1.4880000000000000000, 1.9498560000000000000, 4.6017117724688587423, 31.882740676879848796] So far we considered the Solitaire game. Now suppose that two players take t\ urns, and the first player to reach the goal of having positive capital is declared the winner. Assuming that th\ ey each are give at most, 300, turns the probability of the first-to-move player winning is 0.82529378619195992418 This ends Proposition No., 27, that took, 1.400, seconds to generate. --------------------------------------------- For the set, {-3, 4} We have the following propositions --------------------------------------------- Proposition Number, 28 Consider a casino with a strange roulette, where there were, 2, possible outcomes as follows. Regarding losing You lose, 3, dollars with probability, 1/2 Regarding winning You win, 4, dollars with probability, 1/2 You start with a capital of 0 dollars, and have unlimited credit. Your goal \ in life is to gain a POSITIVE amount, and you quit as soon as you reach that goal. Note that the expected gain in ONE round is, 1/2 Since it is positive, sooner or later you will reach your goal. The probability that the goal is reached within, 300, rounds is , 0.99979171272811865922 The approximate expectation, variance, and scaled moments up to the, 4, -th are [5.5712467055672235279, 228.40971258949175883, 7.5660310179198051087, 83.162310358313187643] Let's compare it to the results of simulating it, 500, times. Of course this changes from time to time and the fraction of the simulated games that for which the goal was reached \ within, 300, rounds is , 1. The crude approximations for the expectation, variance etc. are [5.0080000000000000000, 201.10793600000000000, 8.1109093399082275116, 81.770757714178892479] So far we considered the Solitaire game. Now suppose that two players take t\ urns, and the first player to reach the goal of having positive capital is declared the winner. Assuming that th\ ey each are give at most, 300, turns the probability of the first-to-move player winning is 0.65956741481091082867 This ends Proposition No., 28, that took, 0.220, seconds to generate. --------------------------------------------- For the set, {-3, 1, 3} We have the following propositions --------------------------------------------- Proposition Number, 29 Consider a casino with a strange roulette, where there were, 3, possible outcomes as follows. Regarding losing You lose, 3, dollars with probability, 1/3 Regarding winning You win, 1, dollars with probability, 1/3 You win, 3, dollars with probability, 1/3 You start with a capital of 0 dollars, and have unlimited credit. Your goal \ in life is to gain a POSITIVE amount, and you quit as soon as you reach that goal. Note that the expected gain in ONE round is, 1/3 Since it is positive, sooner or later you will reach your goal. The probability that the goal is reached within, 300, rounds is , 0.99966543725730761487 The approximate expectation, variance, and scaled moments up to the, 4, -th are [5.5730562624699328149, 255.36416824824372966, 7.6321835103082352786, 82.646927583136551990] Let's compare it to the results of simulating it, 500, times. Of course this changes from time to time and the fraction of the simulated games that for which the goal was reached \ within, 300, rounds is , 1. The crude approximations for the expectation, variance etc. are [5.6440000000000000000, 314.95726400000000000, 9.4807378894006946633, 123.00012154080147112] So far we considered the Solitaire game. Now suppose that two players take t\ urns, and the first player to reach the goal of having positive capital is declared the winner. Assuming that th\ ey each are give at most, 300, turns the probability of the first-to-move player winning is 0.73001219055711449712 This ends Proposition No., 29, that took, 1.144, seconds to generate. --------------------------------------------- For the set, {-3, 1, 4} We have the following propositions --------------------------------------------- Proposition Number, 30 Consider a casino with a strange roulette, where there were, 3, possible outcomes as follows. Regarding losing You lose, 3, dollars with probability, 1/3 Regarding winning You win, 1, dollars with probability, 1/3 You win, 4, dollars with probability, 1/3 You start with a capital of 0 dollars, and have unlimited credit. Your goal \ in life is to gain a POSITIVE amount, and you quit as soon as you reach that goal. Note that the expected gain in ONE round is, 2/3 Since it is positive, sooner or later you will reach your goal. The probability that the goal is reached within, 300, rounds is , 0.99999934664818363920 The approximate expectation, variance, and scaled moments up to the, 4, -th are [3.3886890547605176387, 56.774667912733027066, 7.4581118640312331227, 93.487911645220680192] Let's compare it to the results of simulating it, 500, times. Of course this changes from time to time and the fraction of the simulated games that for which the goal was reached \ within, 300, rounds is , 1. The crude approximations for the expectation, variance etc. are [3.2480000000000000000, 34.990496000000000000, 4.8342413274860894741, 32.386256114423075438] So far we considered the Solitaire game. Now suppose that two players take t\ urns, and the first player to reach the goal of having positive capital is declared the winner. Assuming that th\ ey each are give at most, 300, turns the probability of the first-to-move player winning is 0.73033187808517848238 This ends Proposition No., 30, that took, 1.961, seconds to generate. --------------------------------------------- For the set, {-3, 2, 3} We have the following propositions --------------------------------------------- Proposition Number, 31 Consider a casino with a strange roulette, where there were, 3, possible outcomes as follows. Regarding losing You lose, 3, dollars with probability, 1/3 Regarding winning You win, 2, dollars with probability, 1/3 You win, 3, dollars with probability, 1/3 You start with a capital of 0 dollars, and have unlimited credit. Your goal \ in life is to gain a POSITIVE amount, and you quit as soon as you reach that goal. Note that the expected gain in ONE round is, 2/3 Since it is positive, sooner or later you will reach your goal. The probability that the goal is reached within, 300, rounds is , 0.99999981848412008995 The approximate expectation, variance, and scaled moments up to the, 4, -th are [3.4311402338914988888, 48.520037070942614282, 6.9771169635124684611, 83.324306225645992362] Let's compare it to the results of simulating it, 500, times. Of course this changes from time to time and the fraction of the simulated games that for which the goal was reached \ within, 300, rounds is , 1. The crude approximations for the expectation, variance etc. are [3.7660000000000000000, 59.627244000000000000, 5.1081168218851672935, 34.998866445962598118] So far we considered the Solitaire game. Now suppose that two players take t\ urns, and the first player to reach the goal of having positive capital is declared the winner. Assuming that th\ ey each are give at most, 300, turns the probability of the first-to-move player winning is 0.73545051367366735347 This ends Proposition No., 31, that took, 1.914, seconds to generate. --------------------------------------------- For the set, {-3, 2, 4} We have the following propositions --------------------------------------------- Proposition Number, 32 Consider a casino with a strange roulette, where there were, 3, possible outcomes as follows. Regarding losing You lose, 3, dollars with probability, 1/3 Regarding winning You win, 2, dollars with probability, 1/3 You win, 4, dollars with probability, 1/3 You start with a capital of 0 dollars, and have unlimited credit. Your goal \ in life is to gain a POSITIVE amount, and you quit as soon as you reach that goal. Note that the expected gain in ONE round is, 1 Since it is positive, sooner or later you will reach your goal. The probability that the goal is reached within, 300, rounds is , 0.99999999993409227501 The approximate expectation, variance, and scaled moments up to the, 4, -th are [2.5800798775531294845, 19.255867907241190841, 6.2691552081543277881, 67.374648466595585186] Let's compare it to the results of simulating it, 500, times. Of course this changes from time to time and the fraction of the simulated games that for which the goal was reached \ within, 300, rounds is , 1. The crude approximations for the expectation, variance etc. are [2.8440000000000000000, 26.603664000000000000, 5.0979940354392767537, 36.182341330676873588] So far we considered the Solitaire game. Now suppose that two players take t\ urns, and the first player to reach the goal of having positive capital is declared the winner. Assuming that th\ ey each are give at most, 300, turns the probability of the first-to-move player winning is 0.73226720619208637562 This ends Proposition No., 32, that took, 1.880, seconds to generate. --------------------------------------------- For the set, {-3, 3, 4} We have the following propositions --------------------------------------------- Proposition Number, 33 Consider a casino with a strange roulette, where there were, 3, possible outcomes as follows. Regarding losing You lose, 3, dollars with probability, 1/3 Regarding winning You win, 3, dollars with probability, 1/3 You win, 4, dollars with probability, 1/3 You start with a capital of 0 dollars, and have unlimited credit. Your goal \ in life is to gain a POSITIVE amount, and you quit as soon as you reach that goal. Note that the expected gain in ONE round is, 4/3 Since it is positive, sooner or later you will reach your goal. The probability that the goal is reached within, 300, rounds is , 0.99999999999999664742 The approximate expectation, variance, and scaled moments up to the, 4, -th are [2.2449641069518085058, 9.8948704166060432842, 5.5856049844345886376, 54.091748900055526763] Let's compare it to the results of simulating it, 500, times. Of course this changes from time to time and the fraction of the simulated games that for which the goal was reached \ within, 300, rounds is , 1. The crude approximations for the expectation, variance etc. are [2.0880000000000000000, 6.8202560000000000000, 4.1071601762628325756, 24.247381229922380517] So far we considered the Solitaire game. Now suppose that two players take t\ urns, and the first player to reach the goal of having positive capital is declared the winner. Assuming that th\ ey each are give at most, 300, turns the probability of the first-to-move player winning is 0.73303232490569283006 This ends Proposition No., 33, that took, 1.787, seconds to generate. --------------------------------------------- For the set, {-3, 1, 2, 3} We have the following propositions --------------------------------------------- Proposition Number, 34 Consider a casino with a strange roulette, where there were, 4, possible outcomes as follows. Regarding losing You lose, 3, dollars with probability, 1/4 Regarding winning You win, 1, dollars with probability, 1/4 You win, 2, dollars with probability, 1/4 You win, 3, dollars with probability, 1/4 You start with a capital of 0 dollars, and have unlimited credit. Your goal \ in life is to gain a POSITIVE amount, and you quit as soon as you reach that goal. Note that the expected gain in ONE round is, 3/4 Since it is positive, sooner or later you will reach your goal. The probability that the goal is reached within, 300, rounds is , 0.99999999967819757420 The approximate expectation, variance, and scaled moments up to the, 4, -th are [2.5620292896300993978, 21.404156013298988176, 6.5273322445866166450, 73.075841490380458812] Let's compare it to the results of simulating it, 500, times. Of course this changes from time to time and the fraction of the simulated games that for which the goal was reached \ within, 300, rounds is , 1. The crude approximations for the expectation, variance etc. are [2.5560000000000000000, 19.894864000000000000, 5.5413624376806156101, 46.201356595054684116] So far we considered the Solitaire game. Now suppose that two players take t\ urns, and the first player to reach the goal of having positive capital is declared the winner. Assuming that th\ ey each are give at most, 300, turns the probability of the first-to-move player winning is 0.78685673194214267657 This ends Proposition No., 34, that took, 1.691, seconds to generate. --------------------------------------------- For the set, {-3, 1, 2, 4} We have the following propositions --------------------------------------------- Proposition Number, 35 Consider a casino with a strange roulette, where there were, 4, possible outcomes as follows. Regarding losing You lose, 3, dollars with probability, 1/4 Regarding winning You win, 1, dollars with probability, 1/4 You win, 2, dollars with probability, 1/4 You win, 4, dollars with probability, 1/4 You start with a capital of 0 dollars, and have unlimited credit. Your goal \ in life is to gain a POSITIVE amount, and you quit as soon as you reach that goal. Note that the expected gain in ONE round is, 1 Since it is positive, sooner or later you will reach your goal. The probability that the goal is reached within, 300, rounds is , 0.99999999999972815935 The approximate expectation, variance, and scaled moments up to the, 4, -th are [2.1905332927091064283, 12.152948857491995113, 6.0771376963095755272, 62.942812746679042435] Let's compare it to the results of simulating it, 500, times. Of course this changes from time to time and the fraction of the simulated games that for which the goal was reached \ within, 300, rounds is , 1. The crude approximations for the expectation, variance etc. are [2.1100000000000000000, 10.325900000000000000, 5.1593434575498297942, 39.035479727833495043] So far we considered the Solitaire game. Now suppose that two players take t\ urns, and the first player to reach the goal of having positive capital is declared the winner. Assuming that th\ ey each are give at most, 300, turns the probability of the first-to-move player winning is 0.78555785665591081134 This ends Proposition No., 35, that took, 1.616, seconds to generate. --------------------------------------------- For the set, {-3, 1, 3, 4} We have the following propositions --------------------------------------------- Proposition Number, 36 Consider a casino with a strange roulette, where there were, 4, possible outcomes as follows. Regarding losing You lose, 3, dollars with probability, 1/4 Regarding winning You win, 1, dollars with probability, 1/4 You win, 3, dollars with probability, 1/4 You win, 4, dollars with probability, 1/4 You start with a capital of 0 dollars, and have unlimited credit. Your goal \ in life is to gain a POSITIVE amount, and you quit as soon as you reach that goal. Note that the expected gain in ONE round is, 5/4 Since it is positive, sooner or later you will reach your goal. The probability that the goal is reached within, 300, rounds is , 0.99999999999999995684 The approximate expectation, variance, and scaled moments up to the, 4, -th are [1.9663534751688908711, 7.3767276954342133028, 5.6921109961513586010, 55.118854060898566788] Let's compare it to the results of simulating it, 500, times. Of course this changes from time to time and the fraction of the simulated games that for which the goal was reached \ within, 300, rounds is , 1. The crude approximations for the expectation, variance etc. are [1.9640000000000000000, 7.3987040000000000000, 4.9613744452712911034, 37.328787119486949394] So far we considered the Solitaire game. Now suppose that two players take t\ urns, and the first player to reach the goal of having positive capital is declared the winner. Assuming that th\ ey each are give at most, 300, turns the probability of the first-to-move player winning is 0.78701407718634626749 This ends Proposition No., 36, that took, 1.595, seconds to generate. --------------------------------------------- For the set, {-3, 2, 3, 4} We have the following propositions --------------------------------------------- Proposition Number, 37 Consider a casino with a strange roulette, where there were, 4, possible outcomes as follows. Regarding losing You lose, 3, dollars with probability, 1/4 Regarding winning You win, 2, dollars with probability, 1/4 You win, 3, dollars with probability, 1/4 You win, 4, dollars with probability, 1/4 You start with a capital of 0 dollars, and have unlimited credit. Your goal \ in life is to gain a POSITIVE amount, and you quit as soon as you reach that goal. Note that the expected gain in ONE round is, 3/2 Since it is positive, sooner or later you will reach your goal. The probability that the goal is reached within, 300, rounds is , 1.0000000000000000000 The approximate expectation, variance, and scaled moments up to the, 4, -th are [1.8256109930964352531, 4.7464234192785822889, 5.1692741034737513645, 46.147092209931408712] Let's compare it to the results of simulating it, 500, times. Of course this changes from time to time and the fraction of the simulated games that for which the goal was reached \ within, 300, rounds is , 1. The crude approximations for the expectation, variance etc. are [1.8540000000000000000, 5.3846840000000000000, 4.5354999867575330887, 28.566379505594104005] So far we considered the Solitaire game. Now suppose that two players take t\ urns, and the first player to reach the goal of having positive capital is declared the winner. Assuming that th\ ey each are give at most, 300, turns the probability of the first-to-move player winning is 0.78833728257507148473 This ends Proposition No., 37, that took, 1.597, seconds to generate. --------------------------------------------- For the set, {-3, 1, 2, 3, 4} We have the following propositions --------------------------------------------- Proposition Number, 38 Consider a casino with a strange roulette, where there were, 5, possible outcomes as follows. Regarding losing You lose, 3, dollars with probability, 1/5 Regarding winning You win, 1, dollars with probability, 1/5 You win, 2, dollars with probability, 1/5 You win, 3, dollars with probability, 1/5 You win, 4, dollars with probability, 1/5 You start with a capital of 0 dollars, and have unlimited credit. Your goal \ in life is to gain a POSITIVE amount, and you quit as soon as you reach that goal. Note that the expected gain in ONE round is, 7/5 Since it is positive, sooner or later you will reach your goal. The probability that the goal is reached within, 300, rounds is , 1.0000000000000000000 The approximate expectation, variance, and scaled moments up to the, 4, -th are [1.6933482054903195522, 4.0132163661738047910, 5.3373444841493017725, 48.237695626700347538] Let's compare it to the results of simulating it, 500, times. Of course this changes from time to time and the fraction of the simulated games that for which the goal was reached \ within, 300, rounds is , 1. The crude approximations for the expectation, variance etc. are [1.6900000000000000000, 3.5819000000000000000, 3.6451835110176648281, 17.970139705162653446] So far we considered the Solitaire game. Now suppose that two players take t\ urns, and the first player to reach the goal of having positive capital is declared the winner. Assuming that th\ ey each are give at most, 300, turns the probability of the first-to-move player winning is 0.82402535299591684645 This ends Proposition No., 38, that took, 2.094, seconds to generate. --------------------------------------------- For the set, {-4, 1, 4} We have the following propositions --------------------------------------------- Proposition Number, 39 Consider a casino with a strange roulette, where there were, 3, possible outcomes as follows. Regarding losing You lose, 4, dollars with probability, 1/3 Regarding winning You win, 1, dollars with probability, 1/3 You win, 4, dollars with probability, 1/3 You start with a capital of 0 dollars, and have unlimited credit. Your goal \ in life is to gain a POSITIVE amount, and you quit as soon as you reach that goal. Note that the expected gain in ONE round is, 1/3 Since it is positive, sooner or later you will reach your goal. The probability that the goal is reached within, 300, rounds is , 0.99862206752197166967 The approximate expectation, variance, and scaled moments up to the, 4, -th are [6.4767558069449270747, 397.30993785465395161, 7.1891251655283698729, 68.844584955236913128] Let's compare it to the results of simulating it, 500, times. Of course this changes from time to time and the fraction of the simulated games that for which the goal was reached \ within, 300, rounds is , 1. The crude approximations for the expectation, variance etc. are [8.0320000000000000000, 589.31497600000000000, 5.5966303231664266691, 39.975101816880988883] So far we considered the Solitaire game. Now suppose that two players take t\ urns, and the first player to reach the goal of having positive capital is declared the winner. Assuming that th\ ey each are give at most, 300, turns the probability of the first-to-move player winning is 0.72968021132482551347 This ends Proposition No., 39, that took, 2.673, seconds to generate. --------------------------------------------- For the set, {-4, 2, 3} We have the following propositions --------------------------------------------- Proposition Number, 40 Consider a casino with a strange roulette, where there were, 3, possible outcomes as follows. Regarding losing You lose, 4, dollars with probability, 1/3 Regarding winning You win, 2, dollars with probability, 1/3 You win, 3, dollars with probability, 1/3 You start with a capital of 0 dollars, and have unlimited credit. Your goal \ in life is to gain a POSITIVE amount, and you quit as soon as you reach that goal. Note that the expected gain in ONE round is, 1/3 Since it is positive, sooner or later you will reach your goal. The probability that the goal is reached within, 300, rounds is , 0.99895121768782306798 The approximate expectation, variance, and scaled moments up to the, 4, -th are [6.2236286737447089997, 359.94389753549914309, 7.3585489131485863275, 72.904854787173420056] Let's compare it to the results of simulating it, 500, times. Of course this changes from time to time and the fraction of the simulated games that for which the goal was reached \ within, 300, rounds is , 1. The crude approximations for the expectation, variance etc. are [7.5560000000000000000, 520.21486400000000000, 6.5697188589749756979, 59.635015729355310820] So far we considered the Solitaire game. Now suppose that two players take t\ urns, and the first player to reach the goal of having positive capital is declared the winner. Assuming that th\ ey each are give at most, 300, turns the probability of the first-to-move player winning is 0.72995086849048596470 This ends Proposition No., 40, that took, 2.643, seconds to generate. --------------------------------------------- For the set, {-4, 2, 4} We have the following propositions --------------------------------------------- Proposition Number, 41 Consider a casino with a strange roulette, where there were, 3, possible outcomes as follows. Regarding losing You lose, 4, dollars with probability, 1/3 Regarding winning You win, 2, dollars with probability, 1/3 You win, 4, dollars with probability, 1/3 You start with a capital of 0 dollars, and have unlimited credit. Your goal \ in life is to gain a POSITIVE amount, and you quit as soon as you reach that goal. Note that the expected gain in ONE round is, 2/3 Since it is positive, sooner or later you will reach your goal. The probability that the goal is reached within, 300, rounds is , 0.99998893219203375184 The approximate expectation, variance, and scaled moments up to the, 4, -th are [4.3313585940495090208, 106.69659493243132313, 7.4854954879301997738, 90.774583912023952633] Let's compare it to the results of simulating it, 500, times. Of course this changes from time to time and the fraction of the simulated games that for which the goal was reached \ within, 300, rounds is , 1. The crude approximations for the expectation, variance etc. are [4.0760000000000000000, 59.490224000000000000, 4.5479938645430154619, 29.497703651082552772] So far we considered the Solitaire game. Now suppose that two players take t\ urns, and the first player to reach the goal of having positive capital is declared the winner. Assuming that th\ ey each are give at most, 300, turns the probability of the first-to-move player winning is 0.73000272688246834578 This ends Proposition No., 41, that took, 1.359, seconds to generate. --------------------------------------------- For the set, {-4, 3, 4} We have the following propositions --------------------------------------------- Proposition Number, 42 Consider a casino with a strange roulette, where there were, 3, possible outcomes as follows. Regarding losing You lose, 4, dollars with probability, 1/3 Regarding winning You win, 3, dollars with probability, 1/3 You win, 4, dollars with probability, 1/3 You start with a capital of 0 dollars, and have unlimited credit. Your goal \ in life is to gain a POSITIVE amount, and you quit as soon as you reach that goal. Note that the expected gain in ONE round is, 1 Since it is positive, sooner or later you will reach your goal. The probability that the goal is reached within, 300, rounds is , 0.99999997929337504187 The approximate expectation, variance, and scaled moments up to the, 4, -th are [3.2012329693819086319, 36.127464307272663206, 6.6248023164985897899, 76.093210595878087199] Let's compare it to the results of simulating it, 500, times. Of course this changes from time to time and the fraction of the simulated games that for which the goal was reached \ within, 300, rounds is , 1. The crude approximations for the expectation, variance etc. are [3.7700000000000000000, 52.881100000000000000, 4.8074659830218348284, 30.351498643239612652] So far we considered the Solitaire game. Now suppose that two players take t\ urns, and the first player to reach the goal of having positive capital is declared the winner. Assuming that th\ ey each are give at most, 300, turns the probability of the first-to-move player winning is 0.73624950343304387343 This ends Proposition No., 42, that took, 2.496, seconds to generate. --------------------------------------------- For the set, {-4, 1, 2, 3} We have the following propositions --------------------------------------------- Proposition Number, 43 Consider a casino with a strange roulette, where there were, 4, possible outcomes as follows. Regarding losing You lose, 4, dollars with probability, 1/4 Regarding winning You win, 1, dollars with probability, 1/4 You win, 2, dollars with probability, 1/4 You win, 3, dollars with probability, 1/4 You start with a capital of 0 dollars, and have unlimited credit. Your goal \ in life is to gain a POSITIVE amount, and you quit as soon as you reach that goal. Note that the expected gain in ONE round is, 1/2 Since it is positive, sooner or later you will reach your goal. The probability that the goal is reached within, 300, rounds is , 0.99998112560244307861 The approximate expectation, variance, and scaled moments up to the, 4, -th are [3.8022153800490147851, 102.49756259267442574, 8.4860863336400500985, 113.31860911110129399] Let's compare it to the results of simulating it, 500, times. Of course this changes from time to time and the fraction of the simulated games that for which the goal was reached \ within, 300, rounds is , 1. The crude approximations for the expectation, variance etc. are [4.5360000000000000000, 180.97270400000000000, 9.2244786910150317017, 112.55917595441953519] So far we considered the Solitaire game. Now suppose that two players take t\ urns, and the first player to reach the goal of having positive capital is declared the winner. Assuming that th\ ey each are give at most, 300, turns the probability of the first-to-move player winning is 0.78443817316480791001 This ends Proposition No., 43, that took, 2.122, seconds to generate. --------------------------------------------- For the set, {-4, 1, 2, 4} We have the following propositions --------------------------------------------- Proposition Number, 44 Consider a casino with a strange roulette, where there were, 4, possible outcomes as follows. Regarding losing You lose, 4, dollars with probability, 1/4 Regarding winning You win, 1, dollars with probability, 1/4 You win, 2, dollars with probability, 1/4 You win, 4, dollars with probability, 1/4 You start with a capital of 0 dollars, and have unlimited credit. Your goal \ in life is to gain a POSITIVE amount, and you quit as soon as you reach that goal. Note that the expected gain in ONE round is, 3/4 Since it is positive, sooner or later you will reach your goal. The probability that the goal is reached within, 300, rounds is , 0.99999983974136727916 The approximate expectation, variance, and scaled moments up to the, 4, -th are [3.0049314176342528327, 42.367201118106899768, 7.5355779106396818590, 96.216691139971218179] Let's compare it to the results of simulating it, 500, times. Of course this changes from time to time and the fraction of the simulated games that for which the goal was reached \ within, 300, rounds is , 1. The crude approximations for the expectation, variance etc. are [2.5000000000000000000, 19.994000000000000000, 4.4577290436584987732, 24.804916967738115773] So far we considered the Solitaire game. Now suppose that two players take t\ urns, and the first player to reach the goal of having positive capital is declared the winner. Assuming that th\ ey each are give at most, 300, turns the probability of the first-to-move player winning is 0.78554269412805044011 This ends Proposition No., 44, that took, 2.176, seconds to generate. --------------------------------------------- For the set, {-4, 1, 3, 4} We have the following propositions --------------------------------------------- Proposition Number, 45 Consider a casino with a strange roulette, where there were, 4, possible outcomes as follows. Regarding losing You lose, 4, dollars with probability, 1/4 Regarding winning You win, 1, dollars with probability, 1/4 You win, 3, dollars with probability, 1/4 You win, 4, dollars with probability, 1/4 You start with a capital of 0 dollars, and have unlimited credit. Your goal \ in life is to gain a POSITIVE amount, and you quit as soon as you reach that goal. Note that the expected gain in ONE round is, 1 Since it is positive, sooner or later you will reach your goal. The probability that the goal is reached within, 300, rounds is , 0.99999999955429995568 The approximate expectation, variance, and scaled moments up to the, 4, -th are [2.5616269755387208183, 21.609814110681138975, 6.6220562293256813133, 75.320939071587170548] Let's compare it to the results of simulating it, 500, times. Of course this changes from time to time and the fraction of the simulated games that for which the goal was reached \ within, 300, rounds is , 1. The crude approximations for the expectation, variance etc. are [2.7460000000000000000, 24.513484000000000000, 6.4244909751369247147, 62.051565884269803657] So far we considered the Solitaire game. Now suppose that two players take t\ urns, and the first player to reach the goal of having positive capital is declared the winner. Assuming that th\ ey each are give at most, 300, turns the probability of the first-to-move player winning is 0.78686353490810514242 This ends Proposition No., 45, that took, 2.123, seconds to generate. --------------------------------------------- For the set, {-4, 2, 3, 4} We have the following propositions --------------------------------------------- Proposition Number, 46 Consider a casino with a strange roulette, where there were, 4, possible outcomes as follows. Regarding losing You lose, 4, dollars with probability, 1/4 Regarding winning You win, 2, dollars with probability, 1/4 You win, 3, dollars with probability, 1/4 You win, 4, dollars with probability, 1/4 You start with a capital of 0 dollars, and have unlimited credit. Your goal \ in life is to gain a POSITIVE amount, and you quit as soon as you reach that goal. Note that the expected gain in ONE round is, 5/4 Since it is positive, sooner or later you will reach your goal. The probability that the goal is reached within, 300, rounds is , 0.99999999999973853629 The approximate expectation, variance, and scaled moments up to the, 4, -th are [2.2419554663348241465, 12.056599109189490719, 5.9939109176129886375, 62.151228129128010646] Let's compare it to the results of simulating it, 500, times. Of course this changes from time to time and the fraction of the simulated games that for which the goal was reached \ within, 300, rounds is , 1. The crude approximations for the expectation, variance etc. are [2.1580000000000000000, 8.7050360000000000000, 3.9667288882687710239, 21.978688782314249155] So far we considered the Solitaire game. Now suppose that two players take t\ urns, and the first player to reach the goal of having positive capital is declared the winner. Assuming that th\ ey each are give at most, 300, turns the probability of the first-to-move player winning is 0.79001099384810698669 This ends Proposition No., 46, that took, 2.131, seconds to generate. --------------------------------------------- For the set, {-4, 1, 2, 3, 4} We have the following propositions --------------------------------------------- Proposition Number, 47 Consider a casino with a strange roulette, where there were, 5, possible outcomes as follows. Regarding losing You lose, 4, dollars with probability, 1/5 Regarding winning You win, 1, dollars with probability, 1/5 You win, 2, dollars with probability, 1/5 You win, 3, dollars with probability, 1/5 You win, 4, dollars with probability, 1/5 You start with a capital of 0 dollars, and have unlimited credit. Your goal \ in life is to gain a POSITIVE amount, and you quit as soon as you reach that goal. Note that the expected gain in ONE round is, 6/5 Since it is positive, sooner or later you will reach your goal. The probability that the goal is reached within, 300, rounds is , 0.99999999999998800718 The approximate expectation, variance, and scaled moments up to the, 4, -th are [2.0068676152529300178, 9.3142921860933158558, 6.1087733827141925414, 63.691178754077343915] Let's compare it to the results of simulating it, 500, times. Of course this changes from time to time and the fraction of the simulated games that for which the goal was reached \ within, 300, rounds is , 1. The crude approximations for the expectation, variance etc. are [2.3120000000000000000, 14.990656000000000000, 4.6381146098237235114, 28.227385271848462553] So far we considered the Solitaire game. Now suppose that two players take t\ urns, and the first player to reach the goal of having positive capital is declared the winner. Assuming that th\ ey each are give at most, 300, turns the probability of the first-to-move player winning is 0.82415336784391021317 This ends Proposition No., 47, that took, 2.768, seconds to generate. --------------------------------------------- For the set, {-2, -1, 4} We have the following propositions --------------------------------------------- Proposition Number, 48 Consider a casino with a strange roulette, where there were, 3, possible outcomes as follows. Regarding losing You lose, 1, dollars with probability, 1/3 You lose, 2, dollars with probability, 1/3 Regarding winning You win, 4, dollars with probability, 1/3 You start with a capital of 0 dollars, and have unlimited credit. Your goal \ in life is to gain a POSITIVE amount, and you quit as soon as you reach that goal. Note that the expected gain in ONE round is, 1/3 Since it is positive, sooner or later you will reach your goal. The probability that the goal is reached within, 300, rounds is , 0.99940003529365470066 The approximate expectation, variance, and scaled moments up to the, 4, -th are [8.1309534923840359104, 387.50446346126078577, 6.2653683178571601470, 55.947896844970907945] Let's compare it to the results of simulating it, 500, times. Of course this changes from time to time and the fraction of the simulated games that for which the goal was reached \ within, 300, rounds is , 0.99800000000000000000 The crude approximations for the expectation, variance etc. are [8.4068136272545090180, 398.38159686105678290, 5.6481964311658674699, 40.355081439121494276] So far we considered the Solitaire game. Now suppose that two players take t\ urns, and the first player to reach the goal of having positive capital is declared the winner. Assuming that th\ ey each are give at most, 300, turns the probability of the first-to-move player winning is 0.58927563978091059730 This ends Proposition No., 48, that took, 1.561, seconds to generate. --------------------------------------------- For the set, {-2, -1, 1, 3} We have the following propositions --------------------------------------------- Proposition Number, 49 Consider a casino with a strange roulette, where there were, 4, possible outcomes as follows. Regarding losing You lose, 1, dollars with probability, 1/4 You lose, 2, dollars with probability, 1/4 Regarding winning You win, 1, dollars with probability, 1/4 You win, 3, dollars with probability, 1/4 You start with a capital of 0 dollars, and have unlimited credit. Your goal \ in life is to gain a POSITIVE amount, and you quit as soon as you reach that goal. Note that the expected gain in ONE round is, 1/4 Since it is positive, sooner or later you will reach your goal. The probability that the goal is reached within, 300, rounds is , 0.99952547292460674567 The approximate expectation, variance, and scaled moments up to the, 4, -th are [7.0767287763113222168, 335.50975481633220883, 6.6764827649445048051, 63.559047071324466880] Let's compare it to the results of simulating it, 500, times. Of course this changes from time to time and the fraction of the simulated games that for which the goal was reached \ within, 300, rounds is , 1. The crude approximations for the expectation, variance etc. are [6.6100000000000000000, 282.67790000000000000, 9.0377876389698110592, 119.98235482937575859] So far we considered the Solitaire game. Now suppose that two players take t\ urns, and the first player to reach the goal of having positive capital is declared the winner. Assuming that th\ ey each are give at most, 300, turns the probability of the first-to-move player winning is 0.63771174115410096594 This ends Proposition No., 49, that took, 1.242, seconds to generate. --------------------------------------------- For the set, {-2, -1, 1, 4} We have the following propositions --------------------------------------------- Proposition Number, 50 Consider a casino with a strange roulette, where there were, 4, possible outcomes as follows. Regarding losing You lose, 1, dollars with probability, 1/4 You lose, 2, dollars with probability, 1/4 Regarding winning You win, 1, dollars with probability, 1/4 You win, 4, dollars with probability, 1/4 You start with a capital of 0 dollars, and have unlimited credit. Your goal \ in life is to gain a POSITIVE amount, and you quit as soon as you reach that goal. Note that the expected gain in ONE round is, 1/2 Since it is positive, sooner or later you will reach your goal. The probability that the goal is reached within, 300, rounds is , 0.99999819823022780420 The approximate expectation, variance, and scaled moments up to the, 4, -th are [4.6645438153308943858, 87.215416855674230095, 6.4189679040184598493, 69.981365412196076909] Let's compare it to the results of simulating it, 500, times. Of course this changes from time to time and the fraction of the simulated games that for which the goal was reached \ within, 300, rounds is , 1. The crude approximations for the expectation, variance etc. are [4.0200000000000000000, 54.935600000000000000, 5.1913533502736696260, 39.517161257644707969] So far we considered the Solitaire game. Now suppose that two players take t\ urns, and the first player to reach the goal of having positive capital is declared the winner. Assuming that th\ ey each are give at most, 300, turns the probability of the first-to-move player winning is 0.64028966429996819441 This ends Proposition No., 50, that took, 1.269, seconds to generate. --------------------------------------------- For the set, {-2, -1, 2, 3} We have the following propositions --------------------------------------------- Proposition Number, 51 Consider a casino with a strange roulette, where there were, 4, possible outcomes as follows. Regarding losing You lose, 1, dollars with probability, 1/4 You lose, 2, dollars with probability, 1/4 Regarding winning You win, 2, dollars with probability, 1/4 You win, 3, dollars with probability, 1/4 You start with a capital of 0 dollars, and have unlimited credit. Your goal \ in life is to gain a POSITIVE amount, and you quit as soon as you reach that goal. Note that the expected gain in ONE round is, 1/2 Since it is positive, sooner or later you will reach your goal. The probability that the goal is reached within, 300, rounds is , 0.99999965347175397575 The approximate expectation, variance, and scaled moments up to the, 4, -th are [4.0923941187126671261, 62.646216426747947145, 6.4133315966698832461, 70.579791243739450327] Let's compare it to the results of simulating it, 500, times. Of course this changes from time to time and the fraction of the simulated games that for which the goal was reached \ within, 300, rounds is , 1. The crude approximations for the expectation, variance etc. are [3.7020000000000000000, 35.641196000000000000, 4.8210986180289190500, 35.533852756287857146] So far we considered the Solitaire game. Now suppose that two players take t\ urns, and the first player to reach the goal of having positive capital is declared the winner. Assuming that th\ ey each are give at most, 300, turns the probability of the first-to-move player winning is 0.64710096584686045492 This ends Proposition No., 51, that took, 1.187, seconds to generate. --------------------------------------------- For the set, {-2, -1, 2, 4} We have the following propositions --------------------------------------------- Proposition Number, 52 Consider a casino with a strange roulette, where there were, 4, possible outcomes as follows. Regarding losing You lose, 1, dollars with probability, 1/4 You lose, 2, dollars with probability, 1/4 Regarding winning You win, 2, dollars with probability, 1/4 You win, 4, dollars with probability, 1/4 You start with a capital of 0 dollars, and have unlimited credit. Your goal \ in life is to gain a POSITIVE amount, and you quit as soon as you reach that goal. Note that the expected gain in ONE round is, 3/4 Since it is positive, sooner or later you will reach your goal. The probability that the goal is reached within, 300, rounds is , 0.99999999962098044370 The approximate expectation, variance, and scaled moments up to the, 4, -th are [3.3591269019207774581, 28.824980907812073820, 5.5135249811518180086, 53.255625006772686429] Let's compare it to the results of simulating it, 500, times. Of course this changes from time to time and the fraction of the simulated games that for which the goal was reached \ within, 300, rounds is , 1. The crude approximations for the expectation, variance etc. are [3.2040000000000000000, 20.462384000000000000, 3.8445064648598679226, 22.989413431841736054] So far we considered the Solitaire game. Now suppose that two players take t\ urns, and the first player to reach the goal of having positive capital is declared the winner. Assuming that th\ ey each are give at most, 300, turns the probability of the first-to-move player winning is 0.64852410702575849199 This ends Proposition No., 52, that took, 1.191, seconds to generate. --------------------------------------------- For the set, {-2, -1, 3, 4} We have the following propositions --------------------------------------------- Proposition Number, 53 Consider a casino with a strange roulette, where there were, 4, possible outcomes as follows. Regarding losing You lose, 1, dollars with probability, 1/4 You lose, 2, dollars with probability, 1/4 Regarding winning You win, 3, dollars with probability, 1/4 You win, 4, dollars with probability, 1/4 You start with a capital of 0 dollars, and have unlimited credit. Your goal \ in life is to gain a POSITIVE amount, and you quit as soon as you reach that goal. Note that the expected gain in ONE round is, 1 Since it is positive, sooner or later you will reach your goal. The probability that the goal is reached within, 300, rounds is , 0.99999999999993457882 The approximate expectation, variance, and scaled moments up to the, 4, -th are [2.7551763578222056204, 14.490128604213511839, 5.1038123428471792864, 45.699263051358724379] Let's compare it to the results of simulating it, 500, times. Of course this changes from time to time and the fraction of the simulated games that for which the goal was reached \ within, 300, rounds is , 1. The crude approximations for the expectation, variance etc. are [2.5300000000000000000, 11.853100000000000000, 5.8588819837846087735, 52.109203619154010040] So far we considered the Solitaire game. Now suppose that two players take t\ urns, and the first player to reach the goal of having positive capital is declared the winner. Assuming that th\ ey each are give at most, 300, turns the probability of the first-to-move player winning is 0.66036430319445686290 This ends Proposition No., 53, that took, 1.163, seconds to generate. --------------------------------------------- For the set, {-2, -1, 1, 2, 3} We have the following propositions --------------------------------------------- Proposition Number, 54 Consider a casino with a strange roulette, where there were, 5, possible outcomes as follows. Regarding losing You lose, 1, dollars with probability, 1/5 You lose, 2, dollars with probability, 1/5 Regarding winning You win, 1, dollars with probability, 1/5 You win, 2, dollars with probability, 1/5 You win, 3, dollars with probability, 1/5 You start with a capital of 0 dollars, and have unlimited credit. Your goal \ in life is to gain a POSITIVE amount, and you quit as soon as you reach that goal. Note that the expected gain in ONE round is, 3/5 Since it is positive, sooner or later you will reach your goal. The probability that the goal is reached within, 300, rounds is , 0.99999999974672919737 The approximate expectation, variance, and scaled moments up to the, 4, -th are [3.0392083465337503348, 25.783143640272418597, 5.7496853317169499361, 57.440433331707932696] Let's compare it to the results of simulating it, 500, times. Of course this changes from time to time and the fraction of the simulated games that for which the goal was reached \ within, 300, rounds is , 1. The crude approximations for the expectation, variance etc. are [2.7200000000000000000, 11.717600000000000000, 3.1921007590572491644, 15.964142167849477377] So far we considered the Solitaire game. Now suppose that two players take t\ urns, and the first player to reach the goal of having positive capital is declared the winner. Assuming that th\ ey each are give at most, 300, turns the probability of the first-to-move player winning is 0.69212192660731951158 This ends Proposition No., 54, that took, 1.558, seconds to generate. --------------------------------------------- For the set, {-2, -1, 1, 2, 4} We have the following propositions --------------------------------------------- Proposition Number, 55 Consider a casino with a strange roulette, where there were, 5, possible outcomes as follows. Regarding losing You lose, 1, dollars with probability, 1/5 You lose, 2, dollars with probability, 1/5 Regarding winning You win, 1, dollars with probability, 1/5 You win, 2, dollars with probability, 1/5 You win, 4, dollars with probability, 1/5 You start with a capital of 0 dollars, and have unlimited credit. Your goal \ in life is to gain a POSITIVE amount, and you quit as soon as you reach that goal. Note that the expected gain in ONE round is, 4/5 Since it is positive, sooner or later you will reach your goal. The probability that the goal is reached within, 300, rounds is , 0.99999999999955079772 The approximate expectation, variance, and scaled moments up to the, 4, -th are [2.7419161758948011532, 16.399201477053069404, 5.1913144092784272272, 47.282510000559725360] Let's compare it to the results of simulating it, 500, times. Of course this changes from time to time and the fraction of the simulated games that for which the goal was reached \ within, 300, rounds is , 1. The crude approximations for the expectation, variance etc. are [2.6320000000000000000, 12.508576000000000000, 4.2418791723464552516, 27.476706308310588342] So far we considered the Solitaire game. Now suppose that two players take t\ urns, and the first player to reach the goal of having positive capital is declared the winner. Assuming that th\ ey each are give at most, 300, turns the probability of the first-to-move player winning is 0.69327040528377615264 This ends Proposition No., 55, that took, 1.580, seconds to generate. --------------------------------------------- For the set, {-2, -1, 1, 3, 4} We have the following propositions --------------------------------------------- Proposition Number, 56 Consider a casino with a strange roulette, where there were, 5, possible outcomes as follows. Regarding losing You lose, 1, dollars with probability, 1/5 You lose, 2, dollars with probability, 1/5 Regarding winning You win, 1, dollars with probability, 1/5 You win, 3, dollars with probability, 1/5 You win, 4, dollars with probability, 1/5 You start with a capital of 0 dollars, and have unlimited credit. Your goal \ in life is to gain a POSITIVE amount, and you quit as soon as you reach that goal. Note that the expected gain in ONE round is, 1 Since it is positive, sooner or later you will reach your goal. The probability that the goal is reached within, 300, rounds is , 0.99999999999999986118 The approximate expectation, variance, and scaled moments up to the, 4, -th are [2.4125756039006910483, 10.131833259492293073, 4.8981638212066391221, 42.165985810789218993] Let's compare it to the results of simulating it, 500, times. Of course this changes from time to time and the fraction of the simulated games that for which the goal was reached \ within, 300, rounds is , 1. The crude approximations for the expectation, variance etc. are [2.3240000000000000000, 12.947024000000000000, 7.9608874177511814228, 97.297913044616082737] So far we considered the Solitaire game. Now suppose that two players take t\ urns, and the first player to reach the goal of having positive capital is declared the winner. Assuming that th\ ey each are give at most, 300, turns the probability of the first-to-move player winning is 0.69741257010168204804 This ends Proposition No., 56, that took, 1.562, seconds to generate. --------------------------------------------- For the set, {-2, -1, 2, 3, 4} We have the following propositions --------------------------------------------- Proposition Number, 57 Consider a casino with a strange roulette, where there were, 5, possible outcomes as follows. Regarding losing You lose, 1, dollars with probability, 1/5 You lose, 2, dollars with probability, 1/5 Regarding winning You win, 2, dollars with probability, 1/5 You win, 3, dollars with probability, 1/5 You win, 4, dollars with probability, 1/5 You start with a capital of 0 dollars, and have unlimited credit. Your goal \ in life is to gain a POSITIVE amount, and you quit as soon as you reach that goal. Note that the expected gain in ONE round is, 6/5 Since it is positive, sooner or later you will reach your goal. The probability that the goal is reached within, 300, rounds is , 1.0000000000000000000 The approximate expectation, variance, and scaled moments up to the, 4, -th are [2.1463365489949180284, 6.2311395228316286767, 4.6434352990103066866, 38.010157841206753720] Let's compare it to the results of simulating it, 500, times. Of course this changes from time to time and the fraction of the simulated games that for which the goal was reached \ within, 300, rounds is , 1. The crude approximations for the expectation, variance etc. are [2.3080000000000000000, 7.7531360000000000000, 4.0901488429654350665, 25.480840644382841956] So far we considered the Solitaire game. Now suppose that two players take t\ urns, and the first player to reach the goal of having positive capital is declared the winner. Assuming that th\ ey each are give at most, 300, turns the probability of the first-to-move player winning is 0.70338738600145148620 This ends Proposition No., 57, that took, 1.620, seconds to generate. --------------------------------------------- For the set, {-2, -1, 1, 2, 3, 4} We have the following propositions --------------------------------------------- Proposition Number, 58 Consider a casino with a strange roulette, where there were, 6, possible outcomes as follows. Regarding losing You lose, 1, dollars with probability, 1/6 You lose, 2, dollars with probability, 1/6 Regarding winning You win, 1, dollars with probability, 1/6 You win, 2, dollars with probability, 1/6 You win, 3, dollars with probability, 1/6 You win, 4, dollars with probability, 1/6 You start with a capital of 0 dollars, and have unlimited credit. Your goal \ in life is to gain a POSITIVE amount, and you quit as soon as you reach that goal. Note that the expected gain in ONE round is, 7/6 Since it is positive, sooner or later you will reach your goal. The probability that the goal is reached within, 300, rounds is , 1.0000000000000000000 The approximate expectation, variance, and scaled moments up to the, 4, -th are [1.9791021477987552391, 5.0558041947970443714, 4.5892050421296446623, 36.931526262966707196] Let's compare it to the results of simulating it, 500, times. Of course this changes from time to time and the fraction of the simulated games that for which the goal was reached \ within, 300, rounds is , 1. The crude approximations for the expectation, variance etc. are [1.9780000000000000000, 5.1935160000000000000, 3.8149803870212762788, 21.133749963173908741] So far we considered the Solitaire game. Now suppose that two players take t\ urns, and the first player to reach the goal of having positive capital is declared the winner. Assuming that th\ ey each are give at most, 300, turns the probability of the first-to-move player winning is 0.73568755768370141461 This ends Proposition No., 58, that took, 1.576, seconds to generate. --------------------------------------------- For the set, {-3, -1, 1, 4} We have the following propositions --------------------------------------------- Proposition Number, 59 Consider a casino with a strange roulette, where there were, 4, possible outcomes as follows. Regarding losing You lose, 1, dollars with probability, 1/4 You lose, 3, dollars with probability, 1/4 Regarding winning You win, 1, dollars with probability, 1/4 You win, 4, dollars with probability, 1/4 You start with a capital of 0 dollars, and have unlimited credit. Your goal \ in life is to gain a POSITIVE amount, and you quit as soon as you reach that goal. Note that the expected gain in ONE round is, 1/4 Since it is positive, sooner or later you will reach your goal. The probability that the goal is reached within, 300, rounds is , 0.99795343753725361769 The approximate expectation, variance, and scaled moments up to the, 4, -th are [8.2645309344277997808, 524.53071953659251765, 6.2753724669632977706, 52.659822585665992585] Let's compare it to the results of simulating it, 500, times. Of course this changes from time to time and the fraction of the simulated games that for which the goal was reached \ within, 300, rounds is , 1. The crude approximations for the expectation, variance etc. are [8.4340000000000000000, 438.58164400000000000, 5.6341245602136871157, 42.416587711323956040] So far we considered the Solitaire game. Now suppose that two players take t\ urns, and the first player to reach the goal of having positive capital is declared the winner. Assuming that th\ ey each are give at most, 300, turns the probability of the first-to-move player winning is 0.63747243712103344667 This ends Proposition No., 59, that took, 1.803, seconds to generate. --------------------------------------------- For the set, {-3, -1, 2, 3} We have the following propositions --------------------------------------------- Proposition Number, 60 Consider a casino with a strange roulette, where there were, 4, possible outcomes as follows. Regarding losing You lose, 1, dollars with probability, 1/4 You lose, 3, dollars with probability, 1/4 Regarding winning You win, 2, dollars with probability, 1/4 You win, 3, dollars with probability, 1/4 You start with a capital of 0 dollars, and have unlimited credit. Your goal \ in life is to gain a POSITIVE amount, and you quit as soon as you reach that goal. Note that the expected gain in ONE round is, 1/4 Since it is positive, sooner or later you will reach your goal. The probability that the goal is reached within, 300, rounds is , 0.99855616694830761194 The approximate expectation, variance, and scaled moments up to the, 4, -th are [7.7039912741170861586, 456.54054184171221230, 6.5422955009122842097, 57.834526523098474847] Let's compare it to the results of simulating it, 500, times. Of course this changes from time to time and the fraction of the simulated games that for which the goal was reached \ within, 300, rounds is , 0.99800000000000000000 The crude approximations for the expectation, variance etc. are [7.3607214428857715431, 348.05024076208529283, 8.2045859659121921333, 99.243045616697420184] So far we considered the Solitaire game. Now suppose that two players take t\ urns, and the first player to reach the goal of having positive capital is declared the winner. Assuming that th\ ey each are give at most, 300, turns the probability of the first-to-move player winning is 0.63823391021536058328 This ends Proposition No., 60, that took, 1.792, seconds to generate. --------------------------------------------- For the set, {-3, -1, 2, 4} We have the following propositions --------------------------------------------- Proposition Number, 61 Consider a casino with a strange roulette, where there were, 4, possible outcomes as follows. Regarding losing You lose, 1, dollars with probability, 1/4 You lose, 3, dollars with probability, 1/4 Regarding winning You win, 2, dollars with probability, 1/4 You win, 4, dollars with probability, 1/4 You start with a capital of 0 dollars, and have unlimited credit. Your goal \ in life is to gain a POSITIVE amount, and you quit as soon as you reach that goal. Note that the expected gain in ONE round is, 1/2 Since it is positive, sooner or later you will reach your goal. The probability that the goal is reached within, 300, rounds is , 0.99998069477529646570 The approximate expectation, variance, and scaled moments up to the, 4, -th are [4.9102243979397099996, 128.00002837928712758, 7.2581081930627763471, 84.465301909307259632] Let's compare it to the results of simulating it, 500, times. Of course this changes from time to time and the fraction of the simulated games that for which the goal was reached \ within, 300, rounds is , 1. The crude approximations for the expectation, variance etc. are [4.9020000000000000000, 109.13639600000000000, 6.0861765089161284288, 54.168158946368408120] So far we considered the Solitaire game. Now suppose that two players take t\ urns, and the first player to reach the goal of having positive capital is declared the winner. Assuming that th\ ey each are give at most, 300, turns the probability of the first-to-move player winning is 0.64628054888799127154 This ends Proposition No., 61, that took, 1.797, seconds to generate. --------------------------------------------- For the set, {-3, -1, 3, 4} We have the following propositions --------------------------------------------- Proposition Number, 62 Consider a casino with a strange roulette, where there were, 4, possible outcomes as follows. Regarding losing You lose, 1, dollars with probability, 1/4 You lose, 3, dollars with probability, 1/4 Regarding winning You win, 3, dollars with probability, 1/4 You win, 4, dollars with probability, 1/4 You start with a capital of 0 dollars, and have unlimited credit. Your goal \ in life is to gain a POSITIVE amount, and you quit as soon as you reach that goal. Note that the expected gain in ONE round is, 3/4 Since it is positive, sooner or later you will reach your goal. The probability that the goal is reached within, 300, rounds is , 0.99999992700116167013 The approximate expectation, variance, and scaled moments up to the, 4, -th are [3.7711260287022386637, 48.047615584871593822, 6.3342111186321298173, 69.403569236307627896] Let's compare it to the results of simulating it, 500, times. Of course this changes from time to time and the fraction of the simulated games that for which the goal was reached \ within, 300, rounds is , 1. The crude approximations for the expectation, variance etc. are [3.8900000000000000000, 66.701900000000000000, 7.3674931821032204889, 74.738787228803581466] So far we considered the Solitaire game. Now suppose that two players take t\ urns, and the first player to reach the goal of having positive capital is declared the winner. Assuming that th\ ey each are give at most, 300, turns the probability of the first-to-move player winning is 0.64769156118239856448 This ends Proposition No., 62, that took, 1.696, seconds to generate. --------------------------------------------- For the set, {-3, -1, 1, 2, 3} We have the following propositions --------------------------------------------- Proposition Number, 63 Consider a casino with a strange roulette, where there were, 5, possible outcomes as follows. Regarding losing You lose, 1, dollars with probability, 1/5 You lose, 3, dollars with probability, 1/5 Regarding winning You win, 1, dollars with probability, 1/5 You win, 2, dollars with probability, 1/5 You win, 3, dollars with probability, 1/5 You start with a capital of 0 dollars, and have unlimited credit. Your goal \ in life is to gain a POSITIVE amount, and you quit as soon as you reach that goal. Note that the expected gain in ONE round is, 2/5 Since it is positive, sooner or later you will reach your goal. The probability that the goal is reached within, 300, rounds is , 0.99997950152255275082 The approximate expectation, variance, and scaled moments up to the, 4, -th are [4.6045845438673654099, 123.07454479811610755, 7.5495556727430023107, 90.797856506097978750] Let's compare it to the results of simulating it, 500, times. Of course this changes from time to time and the fraction of the simulated games that for which the goal was reached \ within, 300, rounds is , 1. The crude approximations for the expectation, variance etc. are [4.8280000000000000000, 115.02241600000000000, 5.1935975878534509566, 36.190215949735193560] So far we considered the Solitaire game. Now suppose that two players take t\ urns, and the first player to reach the goal of having positive capital is declared the winner. Assuming that th\ ey each are give at most, 300, turns the probability of the first-to-move player winning is 0.68839971897988439526 This ends Proposition No., 63, that took, 2.305, seconds to generate. --------------------------------------------- For the set, {-3, -1, 1, 2, 4} We have the following propositions --------------------------------------------- Proposition Number, 64 Consider a casino with a strange roulette, where there were, 5, possible outcomes as follows. Regarding losing You lose, 1, dollars with probability, 1/5 You lose, 3, dollars with probability, 1/5 Regarding winning You win, 1, dollars with probability, 1/5 You win, 2, dollars with probability, 1/5 You win, 4, dollars with probability, 1/5 You start with a capital of 0 dollars, and have unlimited credit. Your goal \ in life is to gain a POSITIVE amount, and you quit as soon as you reach that goal. Note that the expected gain in ONE round is, 3/5 Since it is positive, sooner or later you will reach your goal. The probability that the goal is reached within, 300, rounds is , 0.99999978705396097757 The approximate expectation, variance, and scaled moments up to the, 4, -th are [3.5931696586560005358, 52.401914672106146776, 6.7915815743527956765, 78.824854969020641097] Let's compare it to the results of simulating it, 500, times. Of course this changes from time to time and the fraction of the simulated games that for which the goal was reached \ within, 300, rounds is , 1. The crude approximations for the expectation, variance etc. are [3.6320000000000000000, 45.584576000000000000, 4.2288378502549961362, 24.596818449049041212] So far we considered the Solitaire game. Now suppose that two players take t\ urns, and the first player to reach the goal of having positive capital is declared the winner. Assuming that th\ ey each are give at most, 300, turns the probability of the first-to-move player winning is 0.69107602258846306560 This ends Proposition No., 64, that took, 2.355, seconds to generate. --------------------------------------------- For the set, {-3, -1, 1, 3, 4} We have the following propositions --------------------------------------------- Proposition Number, 65 Consider a casino with a strange roulette, where there were, 5, possible outcomes as follows. Regarding losing You lose, 1, dollars with probability, 1/5 You lose, 3, dollars with probability, 1/5 Regarding winning You win, 1, dollars with probability, 1/5 You win, 3, dollars with probability, 1/5 You win, 4, dollars with probability, 1/5 You start with a capital of 0 dollars, and have unlimited credit. Your goal \ in life is to gain a POSITIVE amount, and you quit as soon as you reach that goal. Note that the expected gain in ONE round is, 4/5 Since it is positive, sooner or later you will reach your goal. The probability that the goal is reached within, 300, rounds is , 0.99999999922306772098 The approximate expectation, variance, and scaled moments up to the, 4, -th are [3.0365686661778547398, 27.029132522319309973, 6.0372623596845832700, 63.204231226427173398] Let's compare it to the results of simulating it, 500, times. Of course this changes from time to time and the fraction of the simulated games that for which the goal was reached \ within, 300, rounds is , 1. The crude approximations for the expectation, variance etc. are [3.4380000000000000000, 35.666156000000000000, 4.8506530336138811087, 33.793418096792834485] So far we considered the Solitaire game. Now suppose that two players take t\ urns, and the first player to reach the goal of having positive capital is declared the winner. Assuming that th\ ey each are give at most, 300, turns the probability of the first-to-move player winning is 0.69246451837789046729 This ends Proposition No., 65, that took, 2.259, seconds to generate. --------------------------------------------- For the set, {-3, -1, 2, 3, 4} We have the following propositions --------------------------------------------- Proposition Number, 66 Consider a casino with a strange roulette, where there were, 5, possible outcomes as follows. Regarding losing You lose, 1, dollars with probability, 1/5 You lose, 3, dollars with probability, 1/5 Regarding winning You win, 2, dollars with probability, 1/5 You win, 3, dollars with probability, 1/5 You win, 4, dollars with probability, 1/5 You start with a capital of 0 dollars, and have unlimited credit. Your goal \ in life is to gain a POSITIVE amount, and you quit as soon as you reach that goal. Note that the expected gain in ONE round is, 1 Since it is positive, sooner or later you will reach your goal. The probability that the goal is reached within, 300, rounds is , 0.99999999999941862659 The approximate expectation, variance, and scaled moments up to the, 4, -th are [2.5848438214246405790, 14.792563629477998405, 5.5656304535651797781, 53.947526651921235290] Let's compare it to the results of simulating it, 500, times. Of course this changes from time to time and the fraction of the simulated games that for which the goal was reached \ within, 300, rounds is , 1. The crude approximations for the expectation, variance etc. are [2.3800000000000000000, 7.4196000000000000000, 3.1114370732474131758, 14.182713759701749261] So far we considered the Solitaire game. Now suppose that two players take t\ urns, and the first player to reach the goal of having positive capital is declared the winner. Assuming that th\ ey each are give at most, 300, turns the probability of the first-to-move player winning is 0.69684280145835716953 This ends Proposition No., 66, that took, 2.205, seconds to generate. --------------------------------------------- For the set, {-3, -1, 1, 2, 3, 4} We have the following propositions --------------------------------------------- Proposition Number, 67 Consider a casino with a strange roulette, where there were, 6, possible outcomes as follows. Regarding losing You lose, 1, dollars with probability, 1/6 You lose, 3, dollars with probability, 1/6 Regarding winning You win, 1, dollars with probability, 1/6 You win, 2, dollars with probability, 1/6 You win, 3, dollars with probability, 1/6 You win, 4, dollars with probability, 1/6 You start with a capital of 0 dollars, and have unlimited credit. Your goal \ in life is to gain a POSITIVE amount, and you quit as soon as you reach that goal. Note that the expected gain in ONE round is, 1 Since it is positive, sooner or later you will reach your goal. The probability that the goal is reached within, 300, rounds is , 0.99999999999999132080 The approximate expectation, variance, and scaled moments up to the, 4, -th are [2.3111961579081205501, 10.981772146618353666, 5.4721915509400617976, 51.966143780881505163] Let's compare it to the results of simulating it, 500, times. Of course this changes from time to time and the fraction of the simulated games that for which the goal was reached \ within, 300, rounds is , 1. The crude approximations for the expectation, variance etc. are [2.1820000000000000000, 9.0928760000000000000, 4.3235413982166412765, 25.556038634606758436] So far we considered the Solitaire game. Now suppose that two players take t\ urns, and the first player to reach the goal of having positive capital is declared the winner. Assuming that th\ ey each are give at most, 300, turns the probability of the first-to-move player winning is 0.73238697412295579112 This ends Proposition No., 67, that took, 2.361, seconds to generate. --------------------------------------------- For the set, {-4, -1, 2, 4} We have the following propositions --------------------------------------------- Proposition Number, 68 Consider a casino with a strange roulette, where there were, 4, possible outcomes as follows. Regarding losing You lose, 1, dollars with probability, 1/4 You lose, 4, dollars with probability, 1/4 Regarding winning You win, 2, dollars with probability, 1/4 You win, 4, dollars with probability, 1/4 You start with a capital of 0 dollars, and have unlimited credit. Your goal \ in life is to gain a POSITIVE amount, and you quit as soon as you reach that goal. Note that the expected gain in ONE round is, 1/4 Since it is positive, sooner or later you will reach your goal. The probability that the goal is reached within, 300, rounds is , 0.99657258632644387962 The approximate expectation, variance, and scaled moments up to the, 4, -th are [8.6063366700899181983, 600.08905385808888884, 6.1690473446866912009, 49.801129911677212883] Let's compare it to the results of simulating it, 500, times. Of course this changes from time to time and the fraction of the simulated games that for which the goal was reached \ within, 300, rounds is , 0.99800000000000000000 The crude approximations for the expectation, variance etc. are [8.9979959919839679359, 690.37875349898193180, 6.3981841456619897035, 52.382014146325920379] So far we considered the Solitaire game. Now suppose that two players take t\ urns, and the first player to reach the goal of having positive capital is declared the winner. Assuming that th\ ey each are give at most, 300, turns the probability of the first-to-move player winning is 0.63758861360268481020 This ends Proposition No., 68, that took, 2.341, seconds to generate. --------------------------------------------- For the set, {-4, -1, 3, 4} We have the following propositions --------------------------------------------- Proposition Number, 69 Consider a casino with a strange roulette, where there were, 4, possible outcomes as follows. Regarding losing You lose, 1, dollars with probability, 1/4 You lose, 4, dollars with probability, 1/4 Regarding winning You win, 3, dollars with probability, 1/4 You win, 4, dollars with probability, 1/4 You start with a capital of 0 dollars, and have unlimited credit. Your goal \ in life is to gain a POSITIVE amount, and you quit as soon as you reach that goal. Note that the expected gain in ONE round is, 1/2 Since it is positive, sooner or later you will reach your goal. The probability that the goal is reached within, 300, rounds is , 0.99988275347074724735 The approximate expectation, variance, and scaled moments up to the, 4, -th are [5.7076685402994492269, 202.28389206811357698, 7.3500720556746091479, 81.125806470231223535] Let's compare it to the results of simulating it, 500, times. Of course this changes from time to time and the fraction of the simulated games that for which the goal was reached \ within, 300, rounds is , 1. The crude approximations for the expectation, variance etc. are [6.9200000000000000000, 284.16560000000000000, 5.0518966281566285351, 32.299917488478903327] So far we considered the Solitaire game. Now suppose that two players take t\ urns, and the first player to reach the goal of having positive capital is declared the winner. Assuming that th\ ey each are give at most, 300, turns the probability of the first-to-move player winning is 0.64002609067493321131 This ends Proposition No., 69, that took, 2.286, seconds to generate. --------------------------------------------- For the set, {-4, -1, 1, 2, 3} We have the following propositions --------------------------------------------- Proposition Number, 70 Consider a casino with a strange roulette, where there were, 5, possible outcomes as follows. Regarding losing You lose, 1, dollars with probability, 1/5 You lose, 4, dollars with probability, 1/5 Regarding winning You win, 1, dollars with probability, 1/5 You win, 2, dollars with probability, 1/5 You win, 3, dollars with probability, 1/5 You start with a capital of 0 dollars, and have unlimited credit. Your goal \ in life is to gain a POSITIVE amount, and you quit as soon as you reach that goal. Note that the expected gain in ONE round is, 1/5 Since it is positive, sooner or later you will reach your goal. The probability that the goal is reached within, 300, rounds is , 0.99674254744537507458 The approximate expectation, variance, and scaled moments up to the, 4, -th are [7.6492846466959953112, 544.32928997049337618, 6.5665653798517607838, 55.949342902938278691] Let's compare it to the results of simulating it, 500, times. Of course this changes from time to time and the fraction of the simulated games that for which the goal was reached \ within, 300, rounds is , 1. The crude approximations for the expectation, variance etc. are [8.0560000000000000000, 583.75686400000000000, 5.4007042514488330709, 35.684451795195866212] So far we considered the Solitaire game. Now suppose that two players take t\ urns, and the first player to reach the goal of having positive capital is declared the winner. Assuming that th\ ey each are give at most, 300, turns the probability of the first-to-move player winning is 0.68653923154890587369 This ends Proposition No., 70, that took, 3.163, seconds to generate. --------------------------------------------- For the set, {-4, -1, 1, 2, 4} We have the following propositions --------------------------------------------- Proposition Number, 71 Consider a casino with a strange roulette, where there were, 5, possible outcomes as follows. Regarding losing You lose, 1, dollars with probability, 1/5 You lose, 4, dollars with probability, 1/5 Regarding winning You win, 1, dollars with probability, 1/5 You win, 2, dollars with probability, 1/5 You win, 4, dollars with probability, 1/5 You start with a capital of 0 dollars, and have unlimited credit. Your goal \ in life is to gain a POSITIVE amount, and you quit as soon as you reach that goal. Note that the expected gain in ONE round is, 2/5 Since it is positive, sooner or later you will reach your goal. The probability that the goal is reached within, 300, rounds is , 0.99982029730951334983 The approximate expectation, variance, and scaled moments up to the, 4, -th are [5.4238269899220004029, 215.80263626175955824, 7.6423884172456175121, 85.452302378405875914] Let's compare it to the results of simulating it, 500, times. Of course this changes from time to time and the fraction of the simulated games that for which the goal was reached \ within, 300, rounds is , 1. The crude approximations for the expectation, variance etc. are [5.5980000000000000000, 207.61639600000000000, 6.3585026212466068253, 55.321970165166254598] So far we considered the Solitaire game. Now suppose that two players take t\ urns, and the first player to reach the goal of having positive capital is declared the winner. Assuming that th\ ey each are give at most, 300, turns the probability of the first-to-move player winning is 0.68774346143482164796 This ends Proposition No., 71, that took, 3.047, seconds to generate. --------------------------------------------- For the set, {-4, -1, 1, 3, 4} We have the following propositions --------------------------------------------- Proposition Number, 72 Consider a casino with a strange roulette, where there were, 5, possible outcomes as follows. Regarding losing You lose, 1, dollars with probability, 1/5 You lose, 4, dollars with probability, 1/5 Regarding winning You win, 1, dollars with probability, 1/5 You win, 3, dollars with probability, 1/5 You win, 4, dollars with probability, 1/5 You start with a capital of 0 dollars, and have unlimited credit. Your goal \ in life is to gain a POSITIVE amount, and you quit as soon as you reach that goal. Note that the expected gain in ONE round is, 3/5 Since it is positive, sooner or later you will reach your goal. The probability that the goal is reached within, 300, rounds is , 0.99999558443647559049 The approximate expectation, variance, and scaled moments up to the, 4, -th are [4.1372810294065336525, 86.169574685850093644, 7.4206522277937903327, 91.147680111166494752] Let's compare it to the results of simulating it, 500, times. Of course this changes from time to time and the fraction of the simulated games that for which the goal was reached \ within, 300, rounds is , 1. The crude approximations for the expectation, variance etc. are [4.2420000000000000000, 61.723436000000000000, 4.1145703121569679430, 23.362126437007647358] So far we considered the Solitaire game. Now suppose that two players take t\ urns, and the first player to reach the goal of having positive capital is declared the winner. Assuming that th\ ey each are give at most, 300, turns the probability of the first-to-move player winning is 0.68914791453884643328 This ends Proposition No., 72, that took, 3.050, seconds to generate. --------------------------------------------- For the set, {-4, -1, 2, 3, 4} We have the following propositions --------------------------------------------- Proposition Number, 73 Consider a casino with a strange roulette, where there were, 5, possible outcomes as follows. Regarding losing You lose, 1, dollars with probability, 1/5 You lose, 4, dollars with probability, 1/5 Regarding winning You win, 2, dollars with probability, 1/5 You win, 3, dollars with probability, 1/5 You win, 4, dollars with probability, 1/5 You start with a capital of 0 dollars, and have unlimited credit. Your goal \ in life is to gain a POSITIVE amount, and you quit as soon as you reach that goal. Note that the expected gain in ONE round is, 4/5 Since it is positive, sooner or later you will reach your goal. The probability that the goal is reached within, 300, rounds is , 0.99999996313213718005 The approximate expectation, variance, and scaled moments up to the, 4, -th are [3.2653634744638223321, 38.829663591622524541, 6.7573626896108167775, 78.454903960877969894] Let's compare it to the results of simulating it, 500, times. Of course this changes from time to time and the fraction of the simulated games that for which the goal was reached \ within, 300, rounds is , 1. The crude approximations for the expectation, variance etc. are [3.1820000000000000000, 25.940876000000000000, 4.2351910472659084968, 24.746642445109803085] So far we considered the Solitaire game. Now suppose that two players take t\ urns, and the first player to reach the goal of having positive capital is declared the winner. Assuming that th\ ey each are give at most, 300, turns the probability of the first-to-move player winning is 0.69224413946724833688 This ends Proposition No., 73, that took, 2.936, seconds to generate. --------------------------------------------- For the set, {-4, -1, 1, 2, 3, 4} We have the following propositions --------------------------------------------- Proposition Number, 74 Consider a casino with a strange roulette, where there were, 6, possible outcomes as follows. Regarding losing You lose, 1, dollars with probability, 1/6 You lose, 4, dollars with probability, 1/6 Regarding winning You win, 1, dollars with probability, 1/6 You win, 2, dollars with probability, 1/6 You win, 3, dollars with probability, 1/6 You win, 4, dollars with probability, 1/6 You start with a capital of 0 dollars, and have unlimited credit. Your goal \ in life is to gain a POSITIVE amount, and you quit as soon as you reach that goal. Note that the expected gain in ONE round is, 5/6 Since it is positive, sooner or later you will reach your goal. The probability that the goal is reached within, 300, rounds is , 0.99999999862529512915 The approximate expectation, variance, and scaled moments up to the, 4, -th are [2.8005190290093854862, 25.504366005060782959, 6.5333227591783216987, 73.427156882017276569] Let's compare it to the results of simulating it, 500, times. Of course this changes from time to time and the fraction of the simulated games that for which the goal was reached \ within, 300, rounds is , 1. The crude approximations for the expectation, variance etc. are [2.4660000000000000000, 12.784844000000000000, 3.8231184709513210341, 21.976782338994875774] So far we considered the Solitaire game. Now suppose that two players take t\ urns, and the first player to reach the goal of having positive capital is declared the winner. Assuming that th\ ey each are give at most, 300, turns the probability of the first-to-move player winning is 0.73001875207586443581 This ends Proposition No., 74, that took, 3.072, seconds to generate. --------------------------------------------- For the set, {-3, -2, 2, 4} We have the following propositions --------------------------------------------- Proposition Number, 75 Consider a casino with a strange roulette, where there were, 4, possible outcomes as follows. Regarding losing You lose, 2, dollars with probability, 1/4 You lose, 3, dollars with probability, 1/4 Regarding winning You win, 2, dollars with probability, 1/4 You win, 4, dollars with probability, 1/4 You start with a capital of 0 dollars, and have unlimited credit. Your goal \ in life is to gain a POSITIVE amount, and you quit as soon as you reach that goal. Note that the expected gain in ONE round is, 1/4 Since it is positive, sooner or later you will reach your goal. The probability that the goal is reached within, 300, rounds is , 0.99708712807078584660 The approximate expectation, variance, and scaled moments up to the, 4, -th are [8.5739677655492796925, 581.54968915529854338, 6.1610477985666302465, 50.051533596624213747] Let's compare it to the results of simulating it, 500, times. Of course this changes from time to time and the fraction of the simulated games that for which the goal was reached \ within, 300, rounds is , 1. The crude approximations for the expectation, variance etc. are [7.6240000000000000000, 426.26662400000000000, 6.3336144590781720446, 55.496032790511669088] So far we considered the Solitaire game. Now suppose that two players take t\ urns, and the first player to reach the goal of having positive capital is declared the winner. Assuming that th\ ey each are give at most, 300, turns the probability of the first-to-move player winning is 0.63743695148284522386 This ends Proposition No., 75, that took, 1.860, seconds to generate. --------------------------------------------- For the set, {-3, -2, 3, 4} We have the following propositions --------------------------------------------- Proposition Number, 76 Consider a casino with a strange roulette, where there were, 4, possible outcomes as follows. Regarding losing You lose, 2, dollars with probability, 1/4 You lose, 3, dollars with probability, 1/4 Regarding winning You win, 3, dollars with probability, 1/4 You win, 4, dollars with probability, 1/4 You start with a capital of 0 dollars, and have unlimited credit. Your goal \ in life is to gain a POSITIVE amount, and you quit as soon as you reach that goal. Note that the expected gain in ONE round is, 1/2 Since it is positive, sooner or later you will reach your goal. The probability that the goal is reached within, 300, rounds is , 0.99993086951917796419 The approximate expectation, variance, and scaled moments up to the, 4, -th are [5.3679314717520098496, 174.88728243921220238, 7.3794157994072286720, 83.445971274257883507] Let's compare it to the results of simulating it, 500, times. Of course this changes from time to time and the fraction of the simulated games that for which the goal was reached \ within, 300, rounds is , 1. The crude approximations for the expectation, variance etc. are [4.6420000000000000000, 84.801836000000000000, 4.6830581831608323383, 30.151375037747420191] So far we considered the Solitaire game. Now suppose that two players take t\ urns, and the first player to reach the goal of having positive capital is declared the winner. Assuming that th\ ey each are give at most, 300, turns the probability of the first-to-move player winning is 0.64671984228421762100 This ends Proposition No., 76, that took, 1.848, seconds to generate. --------------------------------------------- For the set, {-3, -2, 1, 2, 3} We have the following propositions --------------------------------------------- Proposition Number, 77 Consider a casino with a strange roulette, where there were, 5, possible outcomes as follows. Regarding losing You lose, 2, dollars with probability, 1/5 You lose, 3, dollars with probability, 1/5 Regarding winning You win, 1, dollars with probability, 1/5 You win, 2, dollars with probability, 1/5 You win, 3, dollars with probability, 1/5 You start with a capital of 0 dollars, and have unlimited credit. Your goal \ in life is to gain a POSITIVE amount, and you quit as soon as you reach that goal. Note that the expected gain in ONE round is, 1/5 Since it is positive, sooner or later you will reach your goal. The probability that the goal is reached within, 300, rounds is , 0.99720061320928895143 The approximate expectation, variance, and scaled moments up to the, 4, -th are [7.8945369852600943001, 543.46163250540019003, 6.4312793177130123836, 54.234085183850633600] Let's compare it to the results of simulating it, 500, times. Of course this changes from time to time and the fraction of the simulated games that for which the goal was reached \ within, 300, rounds is , 1. The crude approximations for the expectation, variance etc. are [8.0160000000000000000, 639.20374400000000000, 5.7339782886323687452, 38.619566173707532222] So far we considered the Solitaire game. Now suppose that two players take t\ urns, and the first player to reach the goal of having positive capital is declared the winner. Assuming that th\ ey each are give at most, 300, turns the probability of the first-to-move player winning is 0.68638654887387961704 This ends Proposition No., 77, that took, 2.466, seconds to generate. --------------------------------------------- For the set, {-3, -2, 1, 2, 4} We have the following propositions --------------------------------------------- Proposition Number, 78 Consider a casino with a strange roulette, where there were, 5, possible outcomes as follows. Regarding losing You lose, 2, dollars with probability, 1/5 You lose, 3, dollars with probability, 1/5 Regarding winning You win, 1, dollars with probability, 1/5 You win, 2, dollars with probability, 1/5 You win, 4, dollars with probability, 1/5 You start with a capital of 0 dollars, and have unlimited credit. Your goal \ in life is to gain a POSITIVE amount, and you quit as soon as you reach that goal. Note that the expected gain in ONE round is, 2/5 Since it is positive, sooner or later you will reach your goal. The probability that the goal is reached within, 300, rounds is , 0.99988830337667974688 The approximate expectation, variance, and scaled moments up to the, 4, -th are [5.3289104730124490917, 194.13599898332515220, 7.5191334820864562264, 84.698482627374416894] Let's compare it to the results of simulating it, 500, times. Of course this changes from time to time and the fraction of the simulated games that for which the goal was reached \ within, 300, rounds is , 0.99800000000000000000 The crude approximations for the expectation, variance etc. are [5.2424849699398797595, 217.20573009746948807, 8.4706422367775357377, 97.228920676410737376] So far we considered the Solitaire game. Now suppose that two players take t\ urns, and the first player to reach the goal of having positive capital is declared the winner. Assuming that th\ ey each are give at most, 300, turns the probability of the first-to-move player winning is 0.68741567778590313635 This ends Proposition No., 78, that took, 2.527, seconds to generate. --------------------------------------------- For the set, {-3, -2, 1, 3, 4} We have the following propositions --------------------------------------------- Proposition Number, 79 Consider a casino with a strange roulette, where there were, 5, possible outcomes as follows. Regarding losing You lose, 2, dollars with probability, 1/5 You lose, 3, dollars with probability, 1/5 Regarding winning You win, 1, dollars with probability, 1/5 You win, 3, dollars with probability, 1/5 You win, 4, dollars with probability, 1/5 You start with a capital of 0 dollars, and have unlimited credit. Your goal \ in life is to gain a POSITIVE amount, and you quit as soon as you reach that goal. Note that the expected gain in ONE round is, 3/5 Since it is positive, sooner or later you will reach your goal. The probability that the goal is reached within, 300, rounds is , 0.99999823239793618933 The approximate expectation, variance, and scaled moments up to the, 4, -th are [3.9016896401812876509, 73.001947168815394867, 7.2357824467419780273, 87.711091431738501035] Let's compare it to the results of simulating it, 500, times. Of course this changes from time to time and the fraction of the simulated games that for which the goal was reached \ within, 300, rounds is , 1. The crude approximations for the expectation, variance etc. are [3.8420000000000000000, 60.013036000000000000, 5.8137170184344875968, 49.014979856507177884] So far we considered the Solitaire game. Now suppose that two players take t\ urns, and the first player to reach the goal of having positive capital is declared the winner. Assuming that th\ ey each are give at most, 300, turns the probability of the first-to-move player winning is 0.69076319151857968238 This ends Proposition No., 79, that took, 2.388, seconds to generate. --------------------------------------------- For the set, {-3, -2, 2, 3, 4} We have the following propositions --------------------------------------------- Proposition Number, 80 Consider a casino with a strange roulette, where there were, 5, possible outcomes as follows. Regarding losing You lose, 2, dollars with probability, 1/5 You lose, 3, dollars with probability, 1/5 Regarding winning You win, 2, dollars with probability, 1/5 You win, 3, dollars with probability, 1/5 You win, 4, dollars with probability, 1/5 You start with a capital of 0 dollars, and have unlimited credit. Your goal \ in life is to gain a POSITIVE amount, and you quit as soon as you reach that goal. Note that the expected gain in ONE round is, 4/5 Since it is positive, sooner or later you will reach your goal. The probability that the goal is reached within, 300, rounds is , 0.99999999127177588074 The approximate expectation, variance, and scaled moments up to the, 4, -th are [3.2059757230849964344, 34.194159821276686610, 6.3776315661869520620, 70.222894982247096186] Let's compare it to the results of simulating it, 500, times. Of course this changes from time to time and the fraction of the simulated games that for which the goal was reached \ within, 300, rounds is , 1. The crude approximations for the expectation, variance etc. are [3.5280000000000000000, 90.029216000000000000, 11.162153189809407137, 166.11376996570058681] So far we considered the Solitaire game. Now suppose that two players take t\ urns, and the first player to reach the goal of having positive capital is declared the winner. Assuming that th\ ey each are give at most, 300, turns the probability of the first-to-move player winning is 0.69200317161046763276 This ends Proposition No., 80, that took, 2.330, seconds to generate. --------------------------------------------- For the set, {-3, -2, 1, 2, 3, 4} We have the following propositions --------------------------------------------- Proposition Number, 81 Consider a casino with a strange roulette, where there were, 6, possible outcomes as follows. Regarding losing You lose, 2, dollars with probability, 1/6 You lose, 3, dollars with probability, 1/6 Regarding winning You win, 1, dollars with probability, 1/6 You win, 2, dollars with probability, 1/6 You win, 3, dollars with probability, 1/6 You win, 4, dollars with probability, 1/6 You start with a capital of 0 dollars, and have unlimited credit. Your goal \ in life is to gain a POSITIVE amount, and you quit as soon as you reach that goal. Note that the expected gain in ONE round is, 5/6 Since it is positive, sooner or later you will reach your goal. The probability that the goal is reached within, 300, rounds is , 0.99999999979905000908 The approximate expectation, variance, and scaled moments up to the, 4, -th are [2.7381653146026383112, 22.299045929332240523, 6.1682049234307906887, 65.578717352594104327] Let's compare it to the results of simulating it, 500, times. Of course this changes from time to time and the fraction of the simulated games that for which the goal was reached \ within, 300, rounds is , 1. The crude approximations for the expectation, variance etc. are [2.5240000000000000000, 13.109424000000000000, 3.6098039864253677452, 18.253632885238518907] So far we considered the Solitaire game. Now suppose that two players take t\ urns, and the first player to reach the goal of having positive capital is declared the winner. Assuming that th\ ey each are give at most, 300, turns the probability of the first-to-move player winning is 0.73005037293160145351 This ends Proposition No., 81, that took, 2.425, seconds to generate. --------------------------------------------- For the set, {-4, -2, 3, 4} We have the following propositions --------------------------------------------- Proposition Number, 82 Consider a casino with a strange roulette, where there were, 4, possible outcomes as follows. Regarding losing You lose, 2, dollars with probability, 1/4 You lose, 4, dollars with probability, 1/4 Regarding winning You win, 3, dollars with probability, 1/4 You win, 4, dollars with probability, 1/4 You start with a capital of 0 dollars, and have unlimited credit. Your goal \ in life is to gain a POSITIVE amount, and you quit as soon as you reach that goal. Note that the expected gain in ONE round is, 1/4 Since it is positive, sooner or later you will reach your goal. The probability that the goal is reached within, 300, rounds is , 0.99548321646277904002 The approximate expectation, variance, and scaled moments up to the, 4, -th are [8.9386064101267803756, 654.74733020349058741, 6.0289815732946001963, 47.159813467408268412] Let's compare it to the results of simulating it, 500, times. Of course this changes from time to time and the fraction of the simulated games that for which the goal was reached \ within, 300, rounds is , 0.98400000000000000000 The crude approximations for the expectation, variance etc. are [8.7581300813008130081, 773.76060463348535924, 6.3871821672325512546, 49.566139465834637254] So far we considered the Solitaire game. Now suppose that two players take t\ urns, and the first player to reach the goal of having positive capital is declared the winner. Assuming that th\ ey each are give at most, 300, turns the probability of the first-to-move player winning is 0.63740223984937683716 This ends Proposition No., 82, that took, 2.474, seconds to generate. --------------------------------------------- For the set, {-4, -2, 1, 2, 4} We have the following propositions --------------------------------------------- Proposition Number, 83 Consider a casino with a strange roulette, where there were, 5, possible outcomes as follows. Regarding losing You lose, 2, dollars with probability, 1/5 You lose, 4, dollars with probability, 1/5 Regarding winning You win, 1, dollars with probability, 1/5 You win, 2, dollars with probability, 1/5 You win, 4, dollars with probability, 1/5 You start with a capital of 0 dollars, and have unlimited credit. Your goal \ in life is to gain a POSITIVE amount, and you quit as soon as you reach that goal. Note that the expected gain in ONE round is, 1/5 Since it is positive, sooner or later you will reach your goal. The probability that the goal is reached within, 300, rounds is , 0.99504090062596199517 The approximate expectation, variance, and scaled moments up to the, 4, -th are [8.4876830752801991791, 644.75860555277620398, 6.1713320503917403495, 49.022908089916961201] Let's compare it to the results of simulating it, 500, times. Of course this changes from time to time and the fraction of the simulated games that for which the goal was reached \ within, 300, rounds is , 0.99200000000000000000 The crude approximations for the expectation, variance etc. are [8.3346774193548387097, 498.70653941207075963, 5.7891494310153003265, 50.022659079581867137] So far we considered the Solitaire game. Now suppose that two players take t\ urns, and the first player to reach the goal of having positive capital is declared the winner. Assuming that th\ ey each are give at most, 300, turns the probability of the first-to-move player winning is 0.68578794984801440440 This ends Proposition No., 83, that took, 3.289, seconds to generate. --------------------------------------------- For the set, {-4, -2, 1, 3, 4} We have the following propositions --------------------------------------------- Proposition Number, 84 Consider a casino with a strange roulette, where there were, 5, possible outcomes as follows. Regarding losing You lose, 2, dollars with probability, 1/5 You lose, 4, dollars with probability, 1/5 Regarding winning You win, 1, dollars with probability, 1/5 You win, 3, dollars with probability, 1/5 You win, 4, dollars with probability, 1/5 You start with a capital of 0 dollars, and have unlimited credit. Your goal \ in life is to gain a POSITIVE amount, and you quit as soon as you reach that goal. Note that the expected gain in ONE round is, 2/5 Since it is positive, sooner or later you will reach your goal. The probability that the goal is reached within, 300, rounds is , 0.99964769386954616713 The approximate expectation, variance, and scaled moments up to the, 4, -th are [5.8445090457794197265, 266.75865564243694556, 7.4676942862296075027, 79.233879258852047300] Let's compare it to the results of simulating it, 500, times. Of course this changes from time to time and the fraction of the simulated games that for which the goal was reached \ within, 300, rounds is , 1. The crude approximations for the expectation, variance etc. are [5.8020000000000000000, 204.87479600000000000, 6.2026994031959536178, 53.962883012811829189] So far we considered the Solitaire game. Now suppose that two players take t\ urns, and the first player to reach the goal of having positive capital is declared the winner. Assuming that th\ ey each are give at most, 300, turns the probability of the first-to-move player winning is 0.68731219181070209451 This ends Proposition No., 84, that took, 3.102, seconds to generate. --------------------------------------------- For the set, {-4, -2, 2, 3, 4} We have the following propositions --------------------------------------------- Proposition Number, 85 Consider a casino with a strange roulette, where there were, 5, possible outcomes as follows. Regarding losing You lose, 2, dollars with probability, 1/5 You lose, 4, dollars with probability, 1/5 Regarding winning You win, 2, dollars with probability, 1/5 You win, 3, dollars with probability, 1/5 You win, 4, dollars with probability, 1/5 You start with a capital of 0 dollars, and have unlimited credit. Your goal \ in life is to gain a POSITIVE amount, and you quit as soon as you reach that goal. Note that the expected gain in ONE round is, 3/5 Since it is positive, sooner or later you will reach your goal. The probability that the goal is reached within, 300, rounds is , 0.99998923967793249454 The approximate expectation, variance, and scaled moments up to the, 4, -th are [4.3452202955643208010, 103.57942559897132685, 7.5884887908207523682, 93.365131085978953159] Let's compare it to the results of simulating it, 500, times. Of course this changes from time to time and the fraction of the simulated games that for which the goal was reached \ within, 300, rounds is , 1. The crude approximations for the expectation, variance etc. are [5.0920000000000000000, 152.38753600000000000, 7.7485725808012269995, 86.498269511462911007] So far we considered the Solitaire game. Now suppose that two players take t\ urns, and the first player to reach the goal of having positive capital is declared the winner. Assuming that th\ ey each are give at most, 300, turns the probability of the first-to-move player winning is 0.68900781150477543806 This ends Proposition No., 85, that took, 3.129, seconds to generate. --------------------------------------------- For the set, {-4, -2, 1, 2, 3, 4} We have the following propositions --------------------------------------------- Proposition Number, 86 Consider a casino with a strange roulette, where there were, 6, possible outcomes as follows. Regarding losing You lose, 2, dollars with probability, 1/6 You lose, 4, dollars with probability, 1/6 Regarding winning You win, 1, dollars with probability, 1/6 You win, 2, dollars with probability, 1/6 You win, 3, dollars with probability, 1/6 You win, 4, dollars with probability, 1/6 You start with a capital of 0 dollars, and have unlimited credit. Your goal \ in life is to gain a POSITIVE amount, and you quit as soon as you reach that goal. Note that the expected gain in ONE round is, 2/3 Since it is positive, sooner or later you will reach your goal. The probability that the goal is reached within, 300, rounds is , 0.99999942521701152174 The approximate expectation, variance, and scaled moments up to the, 4, -th are [3.4600155070811708567, 55.957610153990227869, 7.3689794314778718248, 91.802054345624643023] Let's compare it to the results of simulating it, 500, times. Of course this changes from time to time and the fraction of the simulated games that for which the goal was reached \ within, 300, rounds is , 1. The crude approximations for the expectation, variance etc. are [3.7100000000000000000, 59.129900000000000000, 4.7969318356449102252, 31.974250813222292798] So far we considered the Solitaire game. Now suppose that two players take t\ urns, and the first player to reach the goal of having positive capital is declared the winner. Assuming that th\ ey each are give at most, 300, turns the probability of the first-to-move player winning is 0.72847824512848687664 This ends Proposition No., 86, that took, 3.149, seconds to generate. --------------------------------------------- For the set, {-4, -3, 1, 3, 4} We have the following propositions --------------------------------------------- Proposition Number, 87 Consider a casino with a strange roulette, where there were, 5, possible outcomes as follows. Regarding losing You lose, 3, dollars with probability, 1/5 You lose, 4, dollars with probability, 1/5 Regarding winning You win, 1, dollars with probability, 1/5 You win, 3, dollars with probability, 1/5 You win, 4, dollars with probability, 1/5 You start with a capital of 0 dollars, and have unlimited credit. Your goal \ in life is to gain a POSITIVE amount, and you quit as soon as you reach that goal. Note that the expected gain in ONE round is, 1/5 Since it is positive, sooner or later you will reach your goal. The probability that the goal is reached within, 300, rounds is , 0.99379614163820283345 The approximate expectation, variance, and scaled moments up to the, 4, -th are [8.6310583198803533010, 680.86247388765343536, 6.1071149102536368372, 47.649815511081871168] Let's compare it to the results of simulating it, 500, times. Of course this changes from time to time and the fraction of the simulated games that for which the goal was reached \ within, 300, rounds is , 1. The crude approximations for the expectation, variance etc. are [8.0200000000000000000, 625.15560000000000000, 6.3258053941935949221, 49.031800149908986431] So far we considered the Solitaire game. Now suppose that two players take t\ urns, and the first player to reach the goal of having positive capital is declared the winner. Assuming that th\ ey each are give at most, 300, turns the probability of the first-to-move player winning is 0.68625194618981306418 This ends Proposition No., 87, that took, 3.246, seconds to generate. --------------------------------------------- For the set, {-4, -3, 2, 3, 4} We have the following propositions --------------------------------------------- Proposition Number, 88 Consider a casino with a strange roulette, where there were, 5, possible outcomes as follows. Regarding losing You lose, 3, dollars with probability, 1/5 You lose, 4, dollars with probability, 1/5 Regarding winning You win, 2, dollars with probability, 1/5 You win, 3, dollars with probability, 1/5 You win, 4, dollars with probability, 1/5 You start with a capital of 0 dollars, and have unlimited credit. Your goal \ in life is to gain a POSITIVE amount, and you quit as soon as you reach that goal. Note that the expected gain in ONE round is, 2/5 Since it is positive, sooner or later you will reach your goal. The probability that the goal is reached within, 300, rounds is , 0.99941118161272223531 The approximate expectation, variance, and scaled moments up to the, 4, -th are [6.2884210599968920023, 316.69776483944873721, 7.2620656098368490250, 73.350853479140315624] Let's compare it to the results of simulating it, 500, times. Of course this changes from time to time and the fraction of the simulated games that for which the goal was reached \ within, 300, rounds is , 1. The crude approximations for the expectation, variance etc. are [6.3520000000000000000, 275.78009600000000000, 7.0247222081686639226, 66.786959499861157605] So far we considered the Solitaire game. Now suppose that two players take t\ urns, and the first player to reach the goal of having positive capital is declared the winner. Assuming that th\ ey each are give at most, 300, turns the probability of the first-to-move player winning is 0.68891746291999231274 This ends Proposition No., 88, that took, 3.241, seconds to generate. --------------------------------------------- For the set, {-4, -3, 1, 2, 3, 4} We have the following propositions --------------------------------------------- Proposition Number, 89 Consider a casino with a strange roulette, where there were, 6, possible outcomes as follows. Regarding losing You lose, 3, dollars with probability, 1/6 You lose, 4, dollars with probability, 1/6 Regarding winning You win, 1, dollars with probability, 1/6 You win, 2, dollars with probability, 1/6 You win, 3, dollars with probability, 1/6 You win, 4, dollars with probability, 1/6 You start with a capital of 0 dollars, and have unlimited credit. Your goal \ in life is to gain a POSITIVE amount, and you quit as soon as you reach that goal. Note that the expected gain in ONE round is, 1/2 Since it is positive, sooner or later you will reach your goal. The probability that the goal is reached within, 300, rounds is , 0.99994530515078709653 The approximate expectation, variance, and scaled moments up to the, 4, -th are [4.5865000319755182045, 147.07025648316123805, 8.0030591513858825805, 97.952897806090378189] Let's compare it to the results of simulating it, 500, times. Of course this changes from time to time and the fraction of the simulated games that for which the goal was reached \ within, 300, rounds is , 1. The crude approximations for the expectation, variance etc. are [5.0440000000000000000, 115.85406400000000000, 5.1308584311083286546, 34.782484272848766286] So far we considered the Solitaire game. Now suppose that two players take t\ urns, and the first player to reach the goal of having positive capital is declared the winner. Assuming that th\ ey each are give at most, 300, turns the probability of the first-to-move player winning is 0.72795853242135371915 This ends Proposition No., 89, that took, 3.254, seconds to generate. --------------------------------------------- For the set, {-3, -2, -1, 3, 4} We have the following propositions --------------------------------------------- Proposition Number, 90 Consider a casino with a strange roulette, where there were, 5, possible outcomes as follows. Regarding losing You lose, 1, dollars with probability, 1/5 You lose, 2, dollars with probability, 1/5 You lose, 3, dollars with probability, 1/5 Regarding winning You win, 3, dollars with probability, 1/5 You win, 4, dollars with probability, 1/5 You start with a capital of 0 dollars, and have unlimited credit. Your goal \ in life is to gain a POSITIVE amount, and you quit as soon as you reach that goal. Note that the expected gain in ONE round is, 1/5 Since it is positive, sooner or later you will reach your goal. The probability that the goal is reached within, 300, rounds is , 0.99434303781479030610 The approximate expectation, variance, and scaled moments up to the, 4, -th are [10.176756126262449628, 759.21186726678999389, 5.5955028785832458823, 40.746618182110460190] Let's compare it to the results of simulating it, 500, times. Of course this changes from time to time and the fraction of the simulated games that for which the goal was reached \ within, 300, rounds is , 0.99600000000000000000 The crude approximations for the expectation, variance etc. are [9.0783132530120481928, 674.04005177335849422, 6.2833268053287368297, 51.387593823935804847] So far we considered the Solitaire game. Now suppose that two players take t\ urns, and the first player to reach the goal of having positive capital is declared the winner. Assuming that th\ ey each are give at most, 300, turns the probability of the first-to-move player winning is 0.60434311342880697709 This ends Proposition No., 90, that took, 2.561, seconds to generate. --------------------------------------------- For the set, {-3, -2, -1, 1, 2, 4} We have the following propositions --------------------------------------------- Proposition Number, 91 Consider a casino with a strange roulette, where there were, 6, possible outcomes as follows. Regarding losing You lose, 1, dollars with probability, 1/6 You lose, 2, dollars with probability, 1/6 You lose, 3, dollars with probability, 1/6 Regarding winning You win, 1, dollars with probability, 1/6 You win, 2, dollars with probability, 1/6 You win, 4, dollars with probability, 1/6 You start with a capital of 0 dollars, and have unlimited credit. Your goal \ in life is to gain a POSITIVE amount, and you quit as soon as you reach that goal. Note that the expected gain in ONE round is, 1/6 Since it is positive, sooner or later you will reach your goal. The probability that the goal is reached within, 300, rounds is , 0.99415425458569915697 The approximate expectation, variance, and scaled moments up to the, 4, -th are [9.6849619678332004181, 740.59670027021065477, 5.7218188568979964013, 42.378893848651144134] Let's compare it to the results of simulating it, 500, times. Of course this changes from time to time and the fraction of the simulated games that for which the goal was reached \ within, 300, rounds is , 0.99400000000000000000 The crude approximations for the expectation, variance etc. are [8.9899396378269617706, 558.27152856778497949, 5.5595247489497081229, 42.688546264783261533] So far we considered the Solitaire game. Now suppose that two players take t\ urns, and the first player to reach the goal of having positive capital is declared the winner. Assuming that th\ ey each are give at most, 300, turns the probability of the first-to-move player winning is 0.63576226988942692664 This ends Proposition No., 91, that took, 2.668, seconds to generate. --------------------------------------------- For the set, {-3, -2, -1, 1, 3, 4} We have the following propositions --------------------------------------------- Proposition Number, 92 Consider a casino with a strange roulette, where there were, 6, possible outcomes as follows. Regarding losing You lose, 1, dollars with probability, 1/6 You lose, 2, dollars with probability, 1/6 You lose, 3, dollars with probability, 1/6 Regarding winning You win, 1, dollars with probability, 1/6 You win, 3, dollars with probability, 1/6 You win, 4, dollars with probability, 1/6 You start with a capital of 0 dollars, and have unlimited credit. Your goal \ in life is to gain a POSITIVE amount, and you quit as soon as you reach that goal. Note that the expected gain in ONE round is, 1/3 Since it is positive, sooner or later you will reach your goal. The probability that the goal is reached within, 300, rounds is , 0.99955055823269941773 The approximate expectation, variance, and scaled moments up to the, 4, -th are [6.7550709066471768970, 316.63094593462197629, 6.8950782167425024184, 67.611307802308239828] Let's compare it to the results of simulating it, 500, times. Of course this changes from time to time and the fraction of the simulated games that for which the goal was reached \ within, 300, rounds is , 1. The crude approximations for the expectation, variance etc. are [8.0340000000000000000, 451.52084400000000000, 6.6603058944591839185, 58.347285977975627679] So far we considered the Solitaire game. Now suppose that two players take t\ urns, and the first player to reach the goal of having positive capital is declared the winner. Assuming that th\ ey each are give at most, 300, turns the probability of the first-to-move player winning is 0.63939646028592453848 This ends Proposition No., 92, that took, 2.514, seconds to generate. --------------------------------------------- For the set, {-3, -2, -1, 2, 3, 4} We have the following propositions --------------------------------------------- Proposition Number, 93 Consider a casino with a strange roulette, where there were, 6, possible outcomes as follows. Regarding losing You lose, 1, dollars with probability, 1/6 You lose, 2, dollars with probability, 1/6 You lose, 3, dollars with probability, 1/6 Regarding winning You win, 2, dollars with probability, 1/6 You win, 3, dollars with probability, 1/6 You win, 4, dollars with probability, 1/6 You start with a capital of 0 dollars, and have unlimited credit. Your goal \ in life is to gain a POSITIVE amount, and you quit as soon as you reach that goal. Note that the expected gain in ONE round is, 1/2 Since it is positive, sooner or later you will reach your goal. The probability that the goal is reached within, 300, rounds is , 0.99998519288628695144 The approximate expectation, variance, and scaled moments up to the, 4, -th are [4.9036102996622079354, 122.10773224775872517, 7.1511967167365451019, 82.764453258344023823] Let's compare it to the results of simulating it, 500, times. Of course this changes from time to time and the fraction of the simulated games that for which the goal was reached \ within, 300, rounds is , 1. The crude approximations for the expectation, variance etc. are [4.8220000000000000000, 87.038316000000000000, 5.2784168274748150041, 39.261304748966563443] So far we considered the Solitaire game. Now suppose that two players take t\ urns, and the first player to reach the goal of having positive capital is declared the winner. Assuming that th\ ey each are give at most, 300, turns the probability of the first-to-move player winning is 0.64362752584310953062 This ends Proposition No., 93, that took, 2.499, seconds to generate. --------------------------------------------- For the set, {-3, -2, -1, 1, 2, 3, 4} We have the following propositions --------------------------------------------- Proposition Number, 94 Consider a casino with a strange roulette, where there were, 7, possible outcomes as follows. Regarding losing You lose, 1, dollars with probability, 1/7 You lose, 2, dollars with probability, 1/7 You lose, 3, dollars with probability, 1/7 Regarding winning You win, 1, dollars with probability, 1/7 You win, 2, dollars with probability, 1/7 You win, 3, dollars with probability, 1/7 You win, 4, dollars with probability, 1/7 You start with a capital of 0 dollars, and have unlimited credit. Your goal \ in life is to gain a POSITIVE amount, and you quit as soon as you reach that goal. Note that the expected gain in ONE round is, 4/7 Since it is positive, sooner or later you will reach your goal. The probability that the goal is reached within, 300, rounds is , 0.99999934830319775939 The approximate expectation, variance, and scaled moments up to the, 4, -th are [3.9056715832461419688, 64.260888711742956066, 6.8384974015825839157, 79.528801145031540781] Let's compare it to the results of simulating it, 500, times. Of course this changes from time to time and the fraction of the simulated games that for which the goal was reached \ within, 300, rounds is , 1. The crude approximations for the expectation, variance etc. are [3.6020000000000000000, 45.263596000000000000, 4.4596398057022571122, 25.486726879860953717] So far we considered the Solitaire game. Now suppose that two players take t\ urns, and the first player to reach the goal of having positive capital is declared the winner. Assuming that th\ ey each are give at most, 300, turns the probability of the first-to-move player winning is 0.67557248467624739638 This ends Proposition No., 94, that took, 2.556, seconds to generate. --------------------------------------------- For the set, {-4, -2, -1, 1, 3, 4} We have the following propositions --------------------------------------------- Proposition Number, 95 Consider a casino with a strange roulette, where there were, 6, possible outcomes as follows. Regarding losing You lose, 1, dollars with probability, 1/6 You lose, 2, dollars with probability, 1/6 You lose, 4, dollars with probability, 1/6 Regarding winning You win, 1, dollars with probability, 1/6 You win, 3, dollars with probability, 1/6 You win, 4, dollars with probability, 1/6 You start with a capital of 0 dollars, and have unlimited credit. Your goal \ in life is to gain a POSITIVE amount, and you quit as soon as you reach that goal. Note that the expected gain in ONE round is, 1/6 Since it is positive, sooner or later you will reach your goal. The probability that the goal is reached within, 300, rounds is , 0.99228857268177227456 The approximate expectation, variance, and scaled moments up to the, 4, -th are [9.7916166340641895818, 780.95210922119495058, 5.6947577382976533927, 41.563709809930969563] Let's compare it to the results of simulating it, 500, times. Of course this changes from time to time and the fraction of the simulated games that for which the goal was reached \ within, 300, rounds is , 0.99000000000000000000 The crude approximations for the expectation, variance etc. are [10.791919191919191919, 986.31427813488419549, 5.8539201014811016447, 43.791809595475619118] So far we considered the Solitaire game. Now suppose that two players take t\ urns, and the first player to reach the goal of having positive capital is declared the winner. Assuming that th\ ey each are give at most, 300, turns the probability of the first-to-move player winning is 0.63598547998491110516 This ends Proposition No., 95, that took, 3.457, seconds to generate. --------------------------------------------- For the set, {-4, -2, -1, 2, 3, 4} We have the following propositions --------------------------------------------- Proposition Number, 96 Consider a casino with a strange roulette, where there were, 6, possible outcomes as follows. Regarding losing You lose, 1, dollars with probability, 1/6 You lose, 2, dollars with probability, 1/6 You lose, 4, dollars with probability, 1/6 Regarding winning You win, 2, dollars with probability, 1/6 You win, 3, dollars with probability, 1/6 You win, 4, dollars with probability, 1/6 You start with a capital of 0 dollars, and have unlimited credit. Your goal \ in life is to gain a POSITIVE amount, and you quit as soon as you reach that goal. Note that the expected gain in ONE round is, 1/3 Since it is positive, sooner or later you will reach your goal. The probability that the goal is reached within, 300, rounds is , 0.99914543425228830978 The approximate expectation, variance, and scaled moments up to the, 4, -th are [7.0931431278858306423, 378.14414849034364207, 6.8330589806712427101, 64.348976594652533903] Let's compare it to the results of simulating it, 500, times. Of course this changes from time to time and the fraction of the simulated games that for which the goal was reached \ within, 300, rounds is , 1. The crude approximations for the expectation, variance etc. are [7.3220000000000000000, 279.83831600000000000, 5.0663850622185912275, 36.104617964116048502] So far we considered the Solitaire game. Now suppose that two players take t\ urns, and the first player to reach the goal of having positive capital is declared the winner. Assuming that th\ ey each are give at most, 300, turns the probability of the first-to-move player winning is 0.63954285584828687658 This ends Proposition No., 96, that took, 3.304, seconds to generate. --------------------------------------------- For the set, {-4, -2, -1, 1, 2, 3, 4} We have the following propositions --------------------------------------------- Proposition Number, 97 Consider a casino with a strange roulette, where there were, 7, possible outcomes as follows. Regarding losing You lose, 1, dollars with probability, 1/7 You lose, 2, dollars with probability, 1/7 You lose, 4, dollars with probability, 1/7 Regarding winning You win, 1, dollars with probability, 1/7 You win, 2, dollars with probability, 1/7 You win, 3, dollars with probability, 1/7 You win, 4, dollars with probability, 1/7 You start with a capital of 0 dollars, and have unlimited credit. Your goal \ in life is to gain a POSITIVE amount, and you quit as soon as you reach that goal. Note that the expected gain in ONE round is, 3/7 Since it is positive, sooner or later you will reach your goal. The probability that the goal is reached within, 300, rounds is , 0.99991428227154261739 The approximate expectation, variance, and scaled moments up to the, 4, -th are [5.2158322980224957847, 178.48569751022981996, 7.5824140433278690509, 87.043239963351729870] Let's compare it to the results of simulating it, 500, times. Of course this changes from time to time and the fraction of the simulated games that for which the goal was reached \ within, 300, rounds is , 1. The crude approximations for the expectation, variance etc. are [5.2100000000000000000, 238.48190000000000000, 9.4305412388035499104, 119.04052785098356179] So far we considered the Solitaire game. Now suppose that two players take t\ urns, and the first player to reach the goal of having positive capital is declared the winner. Assuming that th\ ey each are give at most, 300, turns the probability of the first-to-move player winning is 0.67326821677128170793 This ends Proposition No., 97, that took, 3.465, seconds to generate. --------------------------------------------- For the set, {-4, -3, -1, 2, 3, 4} We have the following propositions --------------------------------------------- Proposition Number, 98 Consider a casino with a strange roulette, where there were, 6, possible outcomes as follows. Regarding losing You lose, 1, dollars with probability, 1/6 You lose, 3, dollars with probability, 1/6 You lose, 4, dollars with probability, 1/6 Regarding winning You win, 2, dollars with probability, 1/6 You win, 3, dollars with probability, 1/6 You win, 4, dollars with probability, 1/6 You start with a capital of 0 dollars, and have unlimited credit. Your goal \ in life is to gain a POSITIVE amount, and you quit as soon as you reach that goal. Note that the expected gain in ONE round is, 1/6 Since it is positive, sooner or later you will reach your goal. The probability that the goal is reached within, 300, rounds is , 0.99125528546504618515 The approximate expectation, variance, and scaled moments up to the, 4, -th are [9.8358540993514065817, 798.95012541222379511, 5.6808847879481510664, 41.191447175309859705] Let's compare it to the results of simulating it, 500, times. Of course this changes from time to time and the fraction of the simulated games that for which the goal was reached \ within, 300, rounds is , 0.99800000000000000000 The crude approximations for the expectation, variance etc. are [7.0901803607214428858, 363.53294966686880776, 5.1809924566058999592, 33.871062888987862864] So far we considered the Solitaire game. Now suppose that two players take t\ urns, and the first player to reach the goal of having positive capital is declared the winner. Assuming that th\ ey each are give at most, 300, turns the probability of the first-to-move player winning is 0.63615680704699314702 This ends Proposition No., 98, that took, 3.360, seconds to generate. --------------------------------------------- For the set, {-4, -3, -1, 1, 2, 3, 4} We have the following propositions --------------------------------------------- Proposition Number, 99 Consider a casino with a strange roulette, where there were, 7, possible outcomes as follows. Regarding losing You lose, 1, dollars with probability, 1/7 You lose, 3, dollars with probability, 1/7 You lose, 4, dollars with probability, 1/7 Regarding winning You win, 1, dollars with probability, 1/7 You win, 2, dollars with probability, 1/7 You win, 3, dollars with probability, 1/7 You win, 4, dollars with probability, 1/7 You start with a capital of 0 dollars, and have unlimited credit. Your goal \ in life is to gain a POSITIVE amount, and you quit as soon as you reach that goal. Note that the expected gain in ONE round is, 2/7 Since it is positive, sooner or later you will reach your goal. The probability that the goal is reached within, 300, rounds is , 0.99847945323289326680 The approximate expectation, variance, and scaled moments up to the, 4, -th are [7.2411569483067126301, 441.14925934097691461, 6.7697561420292141837, 61.400087775606228056] Let's compare it to the results of simulating it, 500, times. Of course this changes from time to time and the fraction of the simulated games that for which the goal was reached \ within, 300, rounds is , 1. The crude approximations for the expectation, variance etc. are [6.6340000000000000000, 370.58804400000000000, 8.0478054096882512546, 85.672094836393010534] So far we considered the Solitaire game. Now suppose that two players take t\ urns, and the first player to reach the goal of having positive capital is declared the winner. Assuming that th\ ey each are give at most, 300, turns the probability of the first-to-move player winning is 0.67126491816652239832 This ends Proposition No., 99, that took, 3.577, seconds to generate. --------------------------------------------- For the set, {-4, -3, -2, 1, 2, 3, 4} We have the following propositions --------------------------------------------- Proposition Number, 100 Consider a casino with a strange roulette, where there were, 7, possible outcomes as follows. Regarding losing You lose, 2, dollars with probability, 1/7 You lose, 3, dollars with probability, 1/7 You lose, 4, dollars with probability, 1/7 Regarding winning You win, 1, dollars with probability, 1/7 You win, 2, dollars with probability, 1/7 You win, 3, dollars with probability, 1/7 You win, 4, dollars with probability, 1/7 You start with a capital of 0 dollars, and have unlimited credit. Your goal \ in life is to gain a POSITIVE amount, and you quit as soon as you reach that goal. Note that the expected gain in ONE round is, 1/7 Since it is positive, sooner or later you will reach your goal. The probability that the goal is reached within, 300, rounds is , 0.99016992430733561399 The approximate expectation, variance, and scaled moments up to the, 4, -th are [9.4471895962404702748, 791.48164012303596825, 5.7835035582853120143, 42.402699150759210593] Let's compare it to the results of simulating it, 500, times. Of course this changes from time to time and the fraction of the simulated games that for which the goal was reached \ within, 300, rounds is , 0.99000000000000000000 The crude approximations for the expectation, variance etc. are [9.3919191919191919192, 865.40599530660136721, 5.9441851686820227288, 42.888981461336004838] So far we considered the Solitaire game. Now suppose that two players take t\ urns, and the first player to reach the goal of having positive capital is declared the winner. Assuming that th\ ey each are give at most, 300, turns the probability of the first-to-move player winning is 0.66978377239366217797 This ends Proposition No., 100, that took, 3.760, seconds to generate. --------------------------------------------- --------------------------------------------- This concludes this book, that took, 186.124, seconds to generate