Numerical Studties of Games Inspired by Gambler's Ruin with Unlimited Creditwith One Positive and One Negative possible steps of size up to, 10 By Shalosh B. Ekhad In this book we will analyze ALL possible games given by two-element sets of\ steps, one negative, one positive where the sum of the two steps is positive, so that the game is guaranteed t\ o finish, every step is equally likely (hence prob. 1/2) 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, 2, -th are [3.2360679539390012400, 25.969186870570338784] 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] 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, 0.168, 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, 2, -th are [2.3829757679062374913, 7.3303351464613770388] 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] 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, 0.116, 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, 2, -th are [2.1561901553356811691, 4.1379708092392878498] 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] 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, 0.110, seconds to generate. --------------------------------------------- For the set, {-1, 5} We have the following propositions --------------------------------------------- Proposition Number, 4 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, 5, 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, 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, 2, -th are [2.0705043232394492605, 3.0294278966942432295] 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 [2.0960000000000000000, 3.8467840000000000000] 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 0.66659367537025468563 This ends Proposition No., 4, that took, 0.123, seconds to generate. --------------------------------------------- For the set, {-1, 6} We have the following propositions --------------------------------------------- Proposition Number, 5 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, 6, 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, 5/2 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, 2, -th are [2.0333823565252879532, 2.5333646929839498424] 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 [2.0380000000000000000, 2.1485560000000000000] 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 0.66664852738586110488 This ends Proposition No., 5, that took, 0.093, seconds to generate. --------------------------------------------- For the set, {-1, 7} We have the following propositions --------------------------------------------- Proposition Number, 6 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, 7, 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 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, 2, -th are [2.0162018012796575781, 2.2851246601005885840] 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 [1.9720000000000000000, 2.2392160000000000000] 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 0.66666214109682132482 This ends Proposition No., 6, that took, 0.090, seconds to generate. --------------------------------------------- For the set, {-1, 8} We have the following propositions --------------------------------------------- Proposition Number, 7 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, 8, 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, 7/2 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, 2, -th are [2.0079692691259719126, 2.1543156053928838790] 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 [2.0660000000000000000, 2.3416440000000000000] 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 0.66666553603508542865 This ends Proposition No., 7, that took, 0.117, seconds to generate. --------------------------------------------- For the set, {-1, 9} We have the following propositions --------------------------------------------- Proposition Number, 8 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, 9, 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, 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, 2, -th are [2.0039488407836202214, 2.0837902727536897390] 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 [2.0300000000000000000, 2.0171000000000000000] 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 0.66666638406945559316 This ends Proposition No., 8, that took, 0.085, seconds to generate. --------------------------------------------- For the set, {-1, 10} We have the following propositions --------------------------------------------- Proposition Number, 9 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, 10, 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, 9/2 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, 2, -th are [2.0019646626354438337, 2.0454431460491004264] 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 [2.0100000000000000000, 2.0859000000000000000] 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 0.66666659602208102975 This ends Proposition No., 9, that took, 0.087, seconds to generate. --------------------------------------------- For the set, {-2, 3} We have the following propositions --------------------------------------------- Proposition Number, 10 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, 2, -th are [4.4507513685529168264, 100.89775936041966347] 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 [5.3860000000000000000, 129.20900400000000000] 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., 10, that took, 0.200, seconds to generate. --------------------------------------------- For the set, {-2, 4} We have the following propositions --------------------------------------------- Proposition Number, 11 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, 2, -th are [3.2360679539390012400, 25.969186870570338784] 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 [3.1960000000000000000, 23.041584000000000000] 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., 11, that took, 0.193, seconds to generate. --------------------------------------------- For the set, {-2, 5} We have the following propositions --------------------------------------------- Proposition Number, 12 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, 5, 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 , 0.99999999999999958160 The approximate expectation, variance, and scaled moments up to the, 2, -th are [2.4980788092235505105, 10.287890784957245137] 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, 10.605936000000000000] 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 0.66530721217790830272 This ends Proposition No., 12, that took, 0.132, seconds to generate. --------------------------------------------- For the set, {-2, 6} We have the following propositions --------------------------------------------- Proposition Number, 13 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, 6, 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, 2 Since it is positive, sooner or later you will reach your goal. The probability that the goal is reached within, 300, rounds is , 0.99999999999999999999 The approximate expectation, variance, and scaled moments up to the, 2, -th are [2.3829757679062374913, 7.3303351464613770388] 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 [2.5720000000000000000, 7.2288160000000000000] 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., 13, that took, 0.126, seconds to generate. --------------------------------------------- For the set, {-2, 7} We have the following propositions --------------------------------------------- Proposition Number, 14 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, 7, 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, 5/2 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, 2, -th are [2.1799782168356385421, 4.6810950101445498996] 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 [2.0740000000000000000, 4.7885240000000000000] 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 0.66635767766409895773 This ends Proposition No., 14, that took, 0.157, seconds to generate. --------------------------------------------- For the set, {-2, 8} 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, 8, 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 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, 2, -th are [2.1561901553356811691, 4.1379708092392878498] 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, 3.7610360000000000000] 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., 15, that took, 0.109, seconds to generate. --------------------------------------------- For the set, {-2, 9} 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, 9, 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, 7/2 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, 2, -th are [2.0765217328530448822, 3.1742895597498094134] 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 [1.9520000000000000000, 2.2056960000000000000] 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 0.66659250148344568031 This ends Proposition No., 16, that took, 0.104, seconds to generate. --------------------------------------------- For the set, {-2, 10} We have the following propositions --------------------------------------------- Proposition Number, 17 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, 10, 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, 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, 2, -th are [2.0705043232394492605, 3.0294278966942432295] 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 [2.1200000000000000000, 3.1696000000000000000] 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 0.66659367537025468563 This ends Proposition No., 17, that took, 0.103, seconds to generate. --------------------------------------------- For the set, {-3, 4} We have the following propositions --------------------------------------------- Proposition Number, 18 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, 2, -th are [5.5712467055672235279, 228.40971258949175883] 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 [5.6760000000000000000, 181.16302400000000000] 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., 18, that took, 0.257, seconds to generate. --------------------------------------------- For the set, {-3, 5} We have the following propositions --------------------------------------------- Proposition Number, 19 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, 5, 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 , 0.99999983602746983012 The approximate expectation, variance, and scaled moments up to the, 2, -th are [3.6716402885026384802, 50.718820779600141675] 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 [4.0460000000000000000, 60.071884000000000000] 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 0.66088610683886991092 This ends Proposition No., 19, that took, 0.174, seconds to generate. --------------------------------------------- For the set, {-3, 6} We have the following propositions --------------------------------------------- Proposition Number, 20 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, 6, 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 , 0.99999999992478859120 The approximate expectation, variance, and scaled moments up to the, 2, -th are [3.2360679539390012400, 25.969186870570338784] 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 [3.2500000000000000000, 22.339500000000000000] 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., 20, that took, 0.154, seconds to generate. --------------------------------------------- For the set, {-3, 7} We have the following propositions --------------------------------------------- Proposition Number, 21 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, 7, 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, 2 Since it is positive, sooner or later you will reach your goal. The probability that the goal is reached within, 300, rounds is , 0.99999999999997908193 The approximate expectation, variance, and scaled moments up to the, 2, -th are [2.5478852470674066669, 11.958510341254403499] 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 [2.7060000000000000000, 14.271564000000000000] 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 0.66528286149665687686 This ends Proposition No., 21, that took, 0.181, seconds to generate. --------------------------------------------- For the set, {-3, 8} We have the following propositions --------------------------------------------- Proposition Number, 22 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, 8, 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, 5/2 Since it is positive, sooner or later you will reach your goal. The probability that the goal is reached within, 300, rounds is , 0.99999999999999999089 The approximate expectation, variance, and scaled moments up to the, 2, -th are [2.4170071477727756433, 8.4443359591733829135] 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 [2.2720000000000000000, 6.8940160000000000000] 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 0.66541425997684352593 This ends Proposition No., 22, that took, 0.124, seconds to generate. --------------------------------------------- For the set, {-3, 9} We have the following propositions --------------------------------------------- Proposition Number, 23 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, 9, 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 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, 2, -th are [2.3829757679062374913, 7.3303351464613770388] 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 [2.4720000000000000000, 9.3772160000000000000] 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., 23, that took, 0.127, seconds to generate. --------------------------------------------- For the set, {-3, 10} We have the following propositions --------------------------------------------- Proposition Number, 24 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, 10, 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, 7/2 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, 2, -th are [2.1869357310844556135, 4.8889182057342132310] 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 [1.9660000000000000000, 2.6208440000000000000] 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 0.66635642425899716711 This ends Proposition No., 24, that took, 0.115, seconds to generate. --------------------------------------------- For the set, {-4, 5} We have the following propositions --------------------------------------------- Proposition Number, 25 Consider a casino with a strange roulette, where there were, 2, possible outcomes as follows. Regarding losing You lose, 4, dollars with probability, 1/2 Regarding winning You win, 5, 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 , 0.99907254114674709884 The approximate expectation, variance, and scaled moments up to the, 2, -th are [6.4519421524669990765, 359.87766934625874178] 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 [7.1880000000000000000, 500.60465600000000000] 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 0.65939937635546403659 This ends Proposition No., 25, that took, 0.296, seconds to generate. --------------------------------------------- For the set, {-4, 6} We have the following propositions --------------------------------------------- Proposition Number, 26 Consider a casino with a strange roulette, where there were, 2, possible outcomes as follows. Regarding losing You lose, 4, dollars with probability, 1/2 Regarding winning You win, 6, 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 , 0.99999262868191959970 The approximate expectation, variance, and scaled moments up to the, 2, -th are [4.4507513685529168264, 100.89775936041966347] 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 [4.2640000000000000000, 103.43830400000000000] 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., 26, that took, 0.187, seconds to generate. --------------------------------------------- For the set, {-4, 7} We have the following propositions --------------------------------------------- Proposition Number, 27 Consider a casino with a strange roulette, where there were, 2, possible outcomes as follows. Regarding losing You lose, 4, dollars with probability, 1/2 Regarding winning You win, 7, 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 , 0.99999997645439126448 The approximate expectation, variance, and scaled moments up to the, 2, -th are [3.4548452555557333015, 39.000497068155423712] 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 [2.9680000000000000000, 14.246976000000000000] 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 0.66103074732479720064 This ends Proposition No., 27, that took, 0.158, seconds to generate. --------------------------------------------- For the set, {-4, 8} 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, 4, dollars with probability, 1/2 Regarding winning You win, 8, 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, 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, 2, -th are [3.2360679539390012400, 25.969186870570338784] 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 [3.7940000000000000000, 38.523564000000000000] 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., 28, that took, 0.219, seconds to generate. --------------------------------------------- For the set, {-4, 9} We have the following propositions --------------------------------------------- Proposition Number, 29 Consider a casino with a strange roulette, where there were, 2, possible outcomes as follows. Regarding losing You lose, 4, dollars with probability, 1/2 Regarding winning You win, 9, 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, 5/2 Since it is positive, sooner or later you will reach your goal. The probability that the goal is reached within, 300, rounds is , 0.99999999999985511700 The approximate expectation, variance, and scaled moments up to the, 2, -th are [2.5736658733491130731, 12.989247373248734043] 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 [2.4080000000000000000, 7.5895360000000000000] 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 0.66527581191702422653 This ends Proposition No., 29, that took, 0.134, seconds to generate. --------------------------------------------- For the set, {-4, 10} We have the following propositions --------------------------------------------- Proposition Number, 30 Consider a casino with a strange roulette, where there were, 2, possible outcomes as follows. Regarding losing You lose, 4, dollars with probability, 1/2 Regarding winning You win, 10, 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 Since it is positive, sooner or later you will reach your goal. The probability that the goal is reached within, 300, rounds is , 0.99999999999999958160 The approximate expectation, variance, and scaled moments up to the, 2, -th are [2.4980788092235505105, 10.287890784957245137] 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 [2.6060000000000000000, 10.766764000000000000] 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 0.66530721217790830272 This ends Proposition No., 30, that took, 0.136, seconds to generate. --------------------------------------------- For the set, {-5, 6} We have the following propositions --------------------------------------------- Proposition Number, 31 Consider a casino with a strange roulette, where there were, 2, possible outcomes as follows. Regarding losing You lose, 5, dollars with probability, 1/2 Regarding winning You win, 6, 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 , 0.99790179481741951784 The approximate expectation, variance, and scaled moments up to the, 2, -th are [7.0670341758736957216, 464.79873867897280570] 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 [6.9038076152304609218, 450.29936425958128682] 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 0.65931324271766174619 This ends Proposition No., 31, that took, 0.314, seconds to generate. --------------------------------------------- For the set, {-5, 7} We have the following propositions --------------------------------------------- Proposition Number, 32 Consider a casino with a strange roulette, where there were, 2, possible outcomes as follows. Regarding losing You lose, 5, dollars with probability, 1/2 Regarding winning You win, 7, 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 , 0.99994490619818313468 The approximate expectation, variance, and scaled moments up to the, 2, -th are [4.8880362419909437434, 152.74595117633286224] 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 [4.7100000000000000000, 135.97790000000000000] 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 0.65989119977901940303 This ends Proposition No., 32, that took, 0.197, seconds to generate. --------------------------------------------- For the set, {-5, 8} We have the following propositions --------------------------------------------- Proposition Number, 33 Consider a casino with a strange roulette, where there were, 2, possible outcomes as follows. Regarding losing You lose, 5, dollars with probability, 1/2 Regarding winning You win, 8, 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 , 0.99999927031455273178 The approximate expectation, variance, and scaled moments up to the, 2, -th are [3.8051388303591487997, 61.363854889002982904] 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 [3.7440000000000000000, 37.246464000000000000] 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 0.66086044000686572610 This ends Proposition No., 33, that took, 0.175, seconds to generate. --------------------------------------------- For the set, {-5, 9} We have the following propositions --------------------------------------------- Proposition Number, 34 Consider a casino with a strange roulette, where there were, 2, possible outcomes as follows. Regarding losing You lose, 5, dollars with probability, 1/2 Regarding winning You win, 9, 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, 2 Since it is positive, sooner or later you will reach your goal. The probability that the goal is reached within, 300, rounds is , 0.99999999282341755738 The approximate expectation, variance, and scaled moments up to the, 2, -th are [3.3631003400806393591, 34.092351578857347412] 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 [3.3320000000000000000, 37.145776000000000000] 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 0.66106679995292341723 This ends Proposition No., 34, that took, 0.166, seconds to generate. --------------------------------------------- For the set, {-5, 10} We have the following propositions --------------------------------------------- Proposition Number, 35 Consider a casino with a strange roulette, where there were, 2, possible outcomes as follows. Regarding losing You lose, 5, dollars with probability, 1/2 Regarding winning You win, 10, 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, 5/2 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, 2, -th are [3.2360679539390012400, 25.969186870570338784] 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 [2.9460000000000000000, 18.919084000000000000] 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., 35, that took, 0.207, seconds to generate. --------------------------------------------- For the set, {-6, 7} We have the following propositions --------------------------------------------- Proposition Number, 36 Consider a casino with a strange roulette, where there were, 2, possible outcomes as follows. Regarding losing You lose, 6, dollars with probability, 1/2 Regarding winning You win, 7, 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 , 0.99649769253634634736 The approximate expectation, variance, and scaled moments up to the, 2, -th are [7.4713498490306759228, 539.38727177371728402] 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 [6.4176706827309236948, 313.72514959436138127] 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 0.65926411595737977418 This ends Proposition No., 36, that took, 0.261, seconds to generate. --------------------------------------------- For the set, {-6, 8} We have the following propositions --------------------------------------------- Proposition Number, 37 Consider a casino with a strange roulette, where there were, 2, possible outcomes as follows. Regarding losing You lose, 6, dollars with probability, 1/2 Regarding winning You win, 8, 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 , 0.99979171272811865922 The approximate expectation, variance, and scaled moments up to the, 2, -th are [5.5712467055672235279, 228.40971258949175883] 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 [6.2680000000000000000, 273.15617600000000000] 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., 37, that took, 0.235, seconds to generate. --------------------------------------------- For the set, {-6, 9} We have the following propositions --------------------------------------------- Proposition Number, 38 Consider a casino with a strange roulette, where there were, 2, possible outcomes as follows. Regarding losing You lose, 6, dollars with probability, 1/2 Regarding winning You win, 9, 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 , 0.99999262868191959970 The approximate expectation, variance, and scaled moments up to the, 2, -th are [4.4507513685529168264, 100.89775936041966347] 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 [4.3540000000000000000, 111.24868400000000000] 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., 38, that took, 0.244, seconds to generate. --------------------------------------------- For the set, {-6, 10} We have the following propositions --------------------------------------------- Proposition Number, 39 Consider a casino with a strange roulette, where there were, 2, possible outcomes as follows. Regarding losing You lose, 6, dollars with probability, 1/2 Regarding winning You win, 10, 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, 2 Since it is positive, sooner or later you will reach your goal. The probability that the goal is reached within, 300, rounds is , 0.99999983602746983012 The approximate expectation, variance, and scaled moments up to the, 2, -th are [3.6716402885026384802, 50.718820779600141675] 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 [3.6340000000000000000, 38.496044000000000000] 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 0.66088610683886991092 This ends Proposition No., 39, that took, 0.173, seconds to generate. --------------------------------------------- For the set, {-7, 8} We have the following propositions --------------------------------------------- Proposition Number, 40 Consider a casino with a strange roulette, where there were, 2, possible outcomes as follows. Regarding losing You lose, 7, dollars with probability, 1/2 Regarding winning You win, 8, 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 , 0.99505736804674418838 The approximate expectation, variance, and scaled moments up to the, 2, -th are [7.7355854945175957030, 591.63246923172007231] 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 [7.6292585170340681363, 529.05493552234729981] 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 0.65923384743612119376 This ends Proposition No., 40, that took, 0.280, seconds to generate. --------------------------------------------- For the set, {-7, 9} We have the following propositions --------------------------------------------- Proposition Number, 41 Consider a casino with a strange roulette, where there were, 2, possible outcomes as follows. Regarding losing You lose, 7, dollars with probability, 1/2 Regarding winning You win, 9, 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 , 0.99951847588210604919 The approximate expectation, variance, and scaled moments up to the, 2, -th are [5.9264346131771250800, 286.99824094712097971] 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 [5.0801603206412825651, 150.53866450335540821] 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 0.65954294316665192167 This ends Proposition No., 41, that took, 0.284, seconds to generate. --------------------------------------------- For the set, {-7, 10} We have the following propositions --------------------------------------------- Proposition Number, 42 Consider a casino with a strange roulette, where there were, 2, possible outcomes as follows. Regarding losing You lose, 7, dollars with probability, 1/2 Regarding winning You win, 10, 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 , 0.99996986084306297945 The approximate expectation, variance, and scaled moments up to the, 2, -th are [4.6863366808911194189, 130.89955578238089748] 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 [4.0860000000000000000, 84.586604000000000000] 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 0.65993418097676632293 This ends Proposition No., 42, that took, 0.185, seconds to generate. --------------------------------------------- For the set, {-8, 9} We have the following propositions --------------------------------------------- Proposition Number, 43 Consider a casino with a strange roulette, where there were, 2, possible outcomes as follows. Regarding losing You lose, 8, dollars with probability, 1/2 Regarding winning You win, 9, 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 , 0.99372493414125209325 The approximate expectation, variance, and scaled moments up to the, 2, -th are [7.9217864754977468822, 632.16541575610801533] 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 [7.1549295774647887324, 488.22348173548332247] 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 0.65921407699742586640 This ends Proposition No., 43, that took, 0.296, seconds to generate. --------------------------------------------- For the set, {-8, 10} We have the following propositions --------------------------------------------- Proposition Number, 44 Consider a casino with a strange roulette, where there were, 2, possible outcomes as follows. Regarding losing You lose, 8, dollars with probability, 1/2 Regarding winning You win, 10, 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 , 0.99907254114674709884 The approximate expectation, variance, and scaled moments up to the, 2, -th are [6.4519421524669990765, 359.87766934625874178] 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 [5.7640000000000000000, 294.49630400000000000] 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 0.65939937635546403659 This ends Proposition No., 44, that took, 0.283, seconds to generate. --------------------------------------------- For the set, {-9, 10} We have the following propositions --------------------------------------------- Proposition Number, 45 Consider a casino with a strange roulette, where there were, 2, possible outcomes as follows. Regarding losing You lose, 9, dollars with probability, 1/2 Regarding winning You win, 10, 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 , 0.99244961569094798481 The approximate expectation, variance, and scaled moments up to the, 2, -th are [8.0345087390658124563, 658.03921489175059144] 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 [8.0080645161290322581, 534.59267689906347555] 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 0.65920055507877253064 This ends Proposition No., 45, that took, 0.324, seconds to generate. --------------------------------------------- ---------------------------------------------