Advice to the risk averse gambler if the Probality Table is [[-1,(i-1)/i],[i\ ,1/i]] for i from 2 to, 15 By Shalosh B. Ekhad ---------------------------------------------------- Suppose you are offered a bet where With probability, 1/2, you lose , 1, dollars . With probability, 1/2, you win , 2, dollars . The expectation , 1/2, is positive, so conventional wisdom tells you that i\ t is a good deal, and you should accept it, but beware, the standard-deviation is, 1.500000000, that should raise a red flag. Indeed, it you are only allowed to play once, then the probability that you \ would lose money is, 1/2 It would be stupid to take this bet. However, by the law of large numbers, and more quantitavely by the Central L\ imit Theorem, if you are allowed to play many times, you can make the p\ robability of losing money as small as you wish (of course, every gamble carries some rish\ ), the question is how many times should you insist on being able to pla\ y? Let p(n) be the probability that if you play n times, you would gain (at lea\ st some) money Thanks to the amazing Almkvist-Zeilberger algorithm, there is a fast way to \ compute this sequence, via a certain recurrence that we do not display of order, 4, and degree of coefficients , 5 Using this recurrence we can compute the following probabilities If you play, 100, times then your probability of winning a positive amount of money is, 0.9995631322 If you play, 200, times then your probability of winning a positive amount of money is, 0.9999991260 If you play, 300, times then your probability of winning a positive amount of money is, 0.9999999661 If you play, 400, times then your probability of winning a positive amount of money is, 0.9999999677 If you play, 500, times then your probability of winning a positive amount of money is, 0.9999999309 If you play, 600, times then your probability of winning a positive amount of money is, 0.9999999201 If you play, 700, times then your probability of winning a positive amount of money is, 0.9999999292 If you play, 800, times then your probability of winning a positive amount of money is, 0.9999999357 If you play, 900, times then your probability of winning a positive amount of money is, 0.9999999176 If you play, 1000, times then your probability of winning a positive amount of money is, 0.9999998791 If you play, 1100, times then your probability of winning a positive amount of money is, 0.9999998121 If you play, 1200, times then your probability of winning a positive amount of money is, 0.9999997704 If you play, 1300, times then your probability of winning a positive amount of money is, 0.9999997677 If you play, 1400, times then your probability of winning a positive amount of money is, 0.9999997379 If you play, 1500, times then your probability of winning a positive amount of money is, 0.9999997044 If you play, 1600, times then your probability of winning a positive amount of money is, 0.9999997379 If you play, 1700, times then your probability of winning a positive amount of money is, 0.9999996992 If you play, 1800, times then your probability of winning a positive amount of money is, 0.9999997045 If you play, 1900, times then your probability of winning a positive amount of money is, 0.9999997344 If you play, 2000, times then your probability of winning a positive amount of money is, 0.9999998022 ------------------------------- Here is a plot of the number of rounds vs. the probability of winding up wit\ h a positive amount of money 1+ HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH +HH +HH +H 0.9+H +H +H +H 0.8+H +H *H *H 0.7*H *H *H *H 0.6*H *H *H *H 0.5**--+---+--+---+---+--+---+---+--+---+---+--+---+---+--+---+---+--+---+---+- 0 500 1000 1500 2000 ------------------------------ This ends this chapter that took, 0.394, seconds to generate. Suppose you are offered a bet where With probability, 2/3, you lose , 1, dollars . With probability, 1/3, you win , 3, dollars . The expectation , 1/3, is positive, so conventional wisdom tells you that i\ t is a good deal, and you should accept it, but beware, the standard-deviation is, 1.885618082, that should raise a red flag. Indeed, it you are only allowed to play once, then the probability that you \ would lose money is, 2/3 It would be stupid to take this bet. However, by the law of large numbers, and more quantitavely by the Central L\ imit Theorem, if you are allowed to play many times, you can make the p\ robability of losing money as small as you wish (of course, every gamble carries some rish\ ), the question is how many times should you insist on being able to pla\ y? Let p(n) be the probability that if you play n times, you would gain (at lea\ st some) money Thanks to the amazing Almkvist-Zeilberger algorithm, there is a fast way to \ compute this sequence, via a certain recurrence that we do not display of order, 5, and degree of coefficients , 9 Using this recurrence we can compute the following probabilities If you play, 100, times then your probability of winning a positive amount of money is, 0.9541716441 If you play, 200, times then your probability of winning a positive amount of money is, 0.9933173514 If you play, 300, times then your probability of winning a positive amount of money is, 0.9989154968 If you play, 400, times then your probability of winning a positive amount of money is, 0.9998154646 If you play, 500, times then your probability of winning a positive amount of money is, 0.9999677313 If you play, 600, times then your probability of winning a positive amount of money is, 0.9999942499 If you play, 700, times then your probability of winning a positive amount of money is, 0.9999989064 If you play, 800, times then your probability of winning a positive amount of money is, 0.9999997347 If you play, 900, times then your probability of winning a positive amount of money is, 0.9999998827 If you play, 1000, times then your probability of winning a positive amount of money is, 0.9999999006 If you play, 1100, times then your probability of winning a positive amount of money is, 0.9999999257 If you play, 1200, times then your probability of winning a positive amount of money is, 0.9999999477 If you play, 1300, times then your probability of winning a positive amount of money is, 0.9999999328 If you play, 1400, times then your probability of winning a positive amount of money is, 0.9999999152 If you play, 1500, times then your probability of winning a positive amount of money is, 0.9999998469 If you play, 1600, times then your probability of winning a positive amount of money is, 0.9999997819 If you play, 1700, times then your probability of winning a positive amount of money is, 0.9999997261 If you play, 1800, times then your probability of winning a positive amount of money is, 0.9999997060 If you play, 1900, times then your probability of winning a positive amount of money is, 0.9999996268 If you play, 2000, times then your probability of winning a positive amount of money is, 0.9999997242 ------------------------------- Here is a plot of the number of rounds vs. the probability of winding up wit\ h a positive amount of money 1+ HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH + HHHH + HHH 0.9+ HH + H +HH 0.8+HH +H +H 0.7+H +H +H 0.6+H *H *H 0.5*H *H *H 0.4*H * -*---+---+--+---+---+--+---+---+--+---+---+--+---+---+--+---+---+--+---+---+- 0 500 1000 1500 2000 ------------------------------ This ends this chapter that took, 0.609, seconds to generate. Suppose you are offered a bet where With probability, 3/4, you lose , 1, dollars . With probability, 1/4, you win , 4, dollars . The expectation , 1/4, is positive, so conventional wisdom tells you that i\ t is a good deal, and you should accept it, but beware, the standard-deviation is, 2.165063510, that should raise a red flag. Indeed, it you are only allowed to play once, then the probability that you \ would lose money is, 3/4 It would be stupid to take this bet. However, by the law of large numbers, and more quantitavely by the Central L\ imit Theorem, if you are allowed to play many times, you can make the p\ robability of losing money as small as you wish (of course, every gamble carries some rish\ ), the question is how many times should you insist on being able to pla\ y? Let p(n) be the probability that if you play n times, you would gain (at lea\ st some) money Thanks to the amazing Almkvist-Zeilberger algorithm, there is a fast way to \ compute this sequence, via a certain recurrence that we do not display of order, 6, and degree of coefficients , 14 Using this recurrence we can compute the following probabilities If you play, 100, times then your probability of winning a positive amount of money is, 0.8511689478 If you play, 200, times then your probability of winning a positive amount of money is, 0.9421502106 If you play, 300, times then your probability of winning a positive amount of money is, 0.9754439137 If you play, 400, times then your probability of winning a positive amount of money is, 0.9891532260 If you play, 500, times then your probability of winning a positive amount of money is, 0.9950964258 If you play, 600, times then your probability of winning a positive amount of money is, 0.9977490275 If you play, 700, times then your probability of winning a positive amount of money is, 0.9989553919 If you play, 800, times then your probability of winning a positive amount of money is, 0.9995112099 If you play, 900, times then your probability of winning a positive amount of money is, 0.9997698214 If you play, 1000, times then your probability of winning a positive amount of money is, 0.9998910766 If you play, 1100, times then your probability of winning a positive amount of money is, 0.9999483451 If you play, 1200, times then your probability of winning a positive amount of money is, 0.9999755134 If you play, 1300, times then your probability of winning a positive amount of money is, 0.9999884507 If you play, 1400, times then your probability of winning a positive amount of money is, 0.9999945927 If you play, 1500, times then your probability of winning a positive amount of money is, 0.9999975687 If you play, 1600, times then your probability of winning a positive amount of money is, 0.9999989910 If you play, 1700, times then your probability of winning a positive amount of money is, 0.9999997342 If you play, 1800, times then your probability of winning a positive amount of money is, 1.000000145 If you play, 1900, times then your probability of winning a positive amount of money is, 1.000000427 If you play, 2000, times then your probability of winning a positive amount of money is, 1.000000563 ------------------------------- Here is a plot of the number of rounds vs. the probability of winding up wit\ h a positive amount of money 1+ HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH + HHHHHH + HHHH 0.9+ HHHH + HH 0.8+ HHH +HHH +HH 0.7+HH +HH +H 0.6+H +H 0.5+H +H *H 0.4*H *H * 0.3* -*---+---+--+---+---+--+---+---+--+---+---+--+---+---+--+---+---+--+---+---+- 0 500 1000 1500 2000 ------------------------------ This ends this chapter that took, 1.420, seconds to generate. Suppose you are offered a bet where With probability, 4/5, you lose , 1, dollars . With probability, 1/5, you win , 5, dollars . The expectation , 1/5, is positive, so conventional wisdom tells you that i\ t is a good deal, and you should accept it, but beware, the standard-deviation is, 2.400000000, that should raise a red flag. Indeed, it you are only allowed to play once, then the probability that you \ would lose money is, 4/5 It would be stupid to take this bet. However, by the law of large numbers, and more quantitavely by the Central L\ imit Theorem, if you are allowed to play many times, you can make the p\ robability of losing money as small as you wish (of course, every gamble carries some rish\ ), the question is how many times should you insist on being able to pla\ y? Let p(n) be the probability that if you play n times, you would gain (at lea\ st some) money Thanks to the amazing Almkvist-Zeilberger algorithm, there is a fast way to \ compute this sequence, via a certain recurrence that we do not display of order, 7, and degree of coefficients , 20 Using this recurrence we can compute the following probabilities If you play, 100, times then your probability of winning a positive amount of money is, 0.8076623972 If you play, 200, times then your probability of winning a positive amount of money is, 0.8761044963 If you play, 300, times then your probability of winning a positive amount of money is, 0.9170368492 If you play, 400, times then your probability of winning a positive amount of money is, 0.9566511400 If you play, 500, times then your probability of winning a positive amount of money is, 0.9695578920 If you play, 600, times then your probability of winning a positive amount of money is, 0.9784815947 If you play, 700, times then your probability of winning a positive amount of money is, 0.9881296604 If you play, 800, times then your probability of winning a positive amount of money is, 0.9914767654 If you play, 900, times then your probability of winning a positive amount of money is, 0.9938645860 If you play, 1000, times then your probability of winning a positive amount of money is, 0.9965371851 If you play, 1100, times then your probability of winning a positive amount of money is, 0.9974868970 If you play, 1200, times then your probability of winning a positive amount of money is, 0.9981737908 If you play, 1300, times then your probability of winning a positive amount of money is, 0.9989558142 If you play, 1400, times then your probability of winning a positive amount of money is, 0.9992372633 If you play, 1500, times then your probability of winning a positive amount of money is, 0.9994423518 If you play, 1600, times then your probability of winning a positive amount of money is, 0.9996781872 If you play, 1700, times then your probability of winning a positive amount of money is, 0.9997637179 If you play, 1800, times then your probability of winning a positive amount of money is, 0.9998263068 If you play, 1900, times then your probability of winning a positive amount of money is, 0.9998987873 If you play, 2000, times then your probability of winning a positive amount of money is, 0.9999252006 ------------------------------- Here is a plot of the number of rounds vs. the probability of winding up wit\ h a positive amount of money 1+ HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH + HHHHHHHHHHH + HHHHHHH + HHHHH + HHHH 0.8+ HHHH + HHHH +HHHH +HHH +HH 0.6+HH +HH +H +H +H 0.4+H *H * * * 0.2*---+---+--+---+---+--+---+---+--+---+---+--+---+---+--+---+---+--+---+---+- 0 500 1000 1500 2000 ------------------------------ This ends this chapter that took, 2.972, seconds to generate. Suppose you are offered a bet where With probability, 5/6, you lose , 1, dollars . With probability, 1/6, you win , 6, dollars . The expectation , 1/6, is positive, so conventional wisdom tells you that i\ t is a good deal, and you should accept it, but beware, the standard-deviation is, 2.608745974, that should raise a red flag. Indeed, it you are only allowed to play once, then the probability that you \ would lose money is, 5/6 It would be stupid to take this bet. However, by the law of large numbers, and more quantitavely by the Central L\ imit Theorem, if you are allowed to play many times, you can make the p\ robability of losing money as small as you wish (of course, every gamble carries some rish\ ), the question is how many times should you insist on being able to pla\ y? Let p(n) be the probability that if you play n times, you would gain (at lea\ st some) money Thanks to the amazing Almkvist-Zeilberger algorithm, there is a fast way to \ compute this sequence, via a certain recurrence that we do not display of order, 8, and degree of coefficients , 27 Using this recurrence we can compute the following probabilities If you play, 100, times then your probability of winning a positive amount of money is, 0.7125790741 If you play, 200, times then your probability of winning a positive amount of money is, 0.8196530507 If you play, 300, times then your probability of winning a positive amount of money is, 0.8787767258 If you play, 400, times then your probability of winning a positive amount of money is, 0.8922193667 If you play, 500, times then your probability of winning a positive amount of money is, 0.9243430746 If you play, 600, times then your probability of winning a positive amount of money is, 0.9461533778 If you play, 700, times then your probability of winning a positive amount of money is, 0.9515696188 If you play, 800, times then your probability of winning a positive amount of money is, 0.9649869323 If you play, 900, times then your probability of winning a positive amount of money is, 0.9745344470 If you play, 1000, times then your probability of winning a positive amount of money is, 0.9813888626 If you play, 1100, times then your probability of winning a positive amount of money is, 0.9831106729 If you play, 1200, times then your probability of winning a positive amount of money is, 0.9875703906 If you play, 1300, times then your probability of winning a positive amount of money is, 0.9908264819 If you play, 1400, times then your probability of winning a positive amount of money is, 0.9916581775 If you play, 1500, times then your probability of winning a positive amount of money is, 0.9938147079 If you play, 1600, times then your probability of winning a positive amount of money is, 0.9954052812 If you play, 1700, times then your probability of winning a positive amount of money is, 0.9965812826 If you play, 1800, times then your probability of winning a positive amount of money is, 0.9968813708 If you play, 1900, times then your probability of winning a positive amount of money is, 0.9976725528 If you play, 2000, times then your probability of winning a positive amount of money is, 0.9982610006 ------------------------------- Here is a plot of the number of rounds vs. the probability of winding up wit\ h a positive amount of money 1+ HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH + HHHHHHHHHHHHHHHHHH + HHHHHHHHHH + HHHHHHHH + HHHHH 0.8+ HHHHHH + HHHHH + HHHH +HHHH +HHH 0.6+HH +HH +H +H 0.4+H +H +H * * 0.2* -*---+---+--+---+---+--+---+---+--+---+---+--+---+---+--+---+---+--+---+---+- 0 500 1000 1500 2000 ------------------------------ This ends this chapter that took, 6.209, seconds to generate. Suppose you are offered a bet where With probability, 6/7, you lose , 1, dollars . With probability, 1/7, you win , 7, dollars . The expectation , 1/7, is positive, so conventional wisdom tells you that i\ t is a good deal, and you should accept it, but beware, the stanadrd-deviation is, 2.799416849, that should raise a red flag. Indeed, it you are only allowed to play once, then the probability that you \ would lose money is, 6/7 It would be stupid to take this bet. However, by the law of large numbers, and more quantitavely by the Central L\ imit Theorem, if you are allowed to play many times, you can make the p\ robability of losing money as small as you wish (of course, every gamble carries some rish\ ), the question is how many times should you insist on being able to pla\ y? Let p(n) be the probability that if you play n times, you would gain (at lea\ st some) money Thanks to the amazing Almkvist-Zeilberger algorithm, there is a fast way to \ compute this sequence, via a certain recurrence that we do not display of order, 9, and degree of coefficients , 35 Using this recurrence we can compute the following probabilities If you play, 100, times then your probability of winning a positive amount of money is, 0.6858890403 If you play, 200, times then your probability of winning a positive amount of money is, 0.7275958586 If you play, 300, times then your probability of winning a positive amount of money is, 0.8105207593 If you play, 400, times then your probability of winning a positive amount of money is, 0.8283845211 If you play, 500, times then your probability of winning a positive amount of money is, 0.8741772121 If you play, 600, times then your probability of winning a positive amount of money is, 0.8845773954 If you play, 700, times then your probability of winning a positive amount of money is, 0.9133564916 If you play, 800, times then your probability of winning a positive amount of money is, 0.9200088259 If you play, 900, times then your probability of winning a positive amount of money is, 0.9391157363 If you play, 1000, times then your probability of winning a positive amount of money is, 0.9435700770 If you play, 1100, times then your probability of winning a positive amount of money is, 0.9566495211 If you play, 1200, times then your probability of winning a positive amount of money is, 0.9597136815 If you play, 1300, times then your probability of winning a positive amount of money is, 0.9688438460 If you play, 1400, times then your probability of winning a positive amount of money is, 0.9709895993 If you play, 1500, times then your probability of winning a positive amount of money is, 0.9774501476 If you play, 1600, times then your probability of winning a positive amount of money is, 0.9789718938 If you play, 1700, times then your probability of winning a positive amount of money is, 0.9835892680 If you play, 1800, times then your probability of winning a positive amount of money is, 0.9846785762 If you play, 1900, times then your probability of winning a positive amount of money is, 0.9880039565 If you play, 2000, times then your probability of winning a positive amount of money is, 0.9887894646 ------------------------------- Here is a plot of the number of rounds vs. the probability of winding up wit\ h a positive amount of money 1+ HHHHHHHHHHHHHHHHHHHHHHHHHHHHH + HHHHHHHHHHHHHHHHHHHHHH + HHHHHHHHHHHHH + HHHHHHHHHH 0.8+ HHHHHHHH + HHHHHHH + HHHHHH + HHHHHH +HHHHH 0.6+HHHH +HHH +HH +HH +H 0.4+H +H +H * * 0.2* -*---+---+--+---+---+--+---+---+--+---+---+--+---+---+--+---+---+--+---+---+- 0 500 1000 1500 2000 ------------------------------ This ends this chapter that took, 12.440, seconds to generate. Suppose you are offered a bet where With probability, 7/8, you lose , 1, dollars . With probability, 1/8, you win , 8, dollars . The expectation , 1/8, is positive, so conventional wisdom tells you that i\ t is a good deal, and you should accept it, but beware, the standard-deviation is, 2.976470225, that should raise a red flag. Indeed, it you are only allowed to play once, then the probability that you \ would lose money is, 7/8 It would be stupid to take this bet. However, by the law of large numbers, and more quantitavely by the Central L\ imit Theorem, if you are allowed to play many times, you can make the p\ robability of losing money as small as you wish (of course, every gamble carries some rish\ ), the question is how many times should you insist on being able to pla\ y? Let p(n) be the probability that if you play n times, you would gain (at lea\ st some) money Thanks to the amazing Almkvist-Zeilberger algorithm, there is a fast way to \ compute this sequence, via a certain recurrence that we do not display of order, 10, and degree of coefficients , 44 Using this recurrence we can compute the following probabilities If you play, 100, times then your probability of winning a positive amount of money is, 0.6052691192 If you play, 200, times then your probability of winning a positive amount of money is, 0.6967221155 If you play, 300, times then your probability of winning a positive amount of money is, 0.7539610451 If you play, 400, times then your probability of winning a positive amount of money is, 0.7955451256 If you play, 500, times then your probability of winning a positive amount of money is, 0.8277013546 If you play, 600, times then your probability of winning a positive amount of money is, 0.8534346727 If you play, 700, times then your probability of winning a positive amount of money is, 0.8744786484 If you play, 800, times then your probability of winning a positive amount of money is, 0.8919459108 If you play, 900, times then your probability of winning a positive amount of money is, 0.8879890179 If you play, 1000, times then your probability of winning a positive amount of money is, 0.9031133319 If you play, 1100, times then your probability of winning a positive amount of money is, 0.9159329851 If you play, 1200, times then your probability of winning a positive amount of money is, 0.9268644812 If you play, 1300, times then your probability of winning a positive amount of money is, 0.9362317251 If you play, 1400, times then your probability of winning a positive amount of money is, 0.9442913672 If you play, 1500, times then your probability of winning a positive amount of money is, 0.9512499059 If you play, 1600, times then your probability of winning a positive amount of money is, 0.9572755186 If you play, 1700, times then your probability of winning a positive amount of money is, 0.9625065890 If you play, 1800, times then your probability of winning a positive amount of money is, 0.9612608469 If you play, 1900, times then your probability of winning a positive amount of money is, 0.9659495874 If you play, 2000, times then your probability of winning a positive amount of money is, 0.9700401764 ------------------------------- Here is a plot of the number of rounds vs. the probability of winding up wit\ h a positive amount of money + HHHHHHHHHHHHHHHHHHHHH + HHHHHHHHHHHHHHHHHHHHHHHH + HHHHHHHHHHHHHHHH + HHHHHHHHHHHH 0.8+ HHHHHHHHHHH + HHHHHHHHHH + HHHHHHHHH +HHHHHHHHH +HHHHHH 0.6+HHHHH +HHH +HH +HH +H 0.4+H +H +H *H 0.2* * -*---+---+--+---+---+--+---+---+--+---+---+--+---+---+--+---+---+--+---+---+- 0 500 1000 1500 2000 ------------------------------ This ends this chapter that took, 20.442, seconds to generate. Suppose you are offered a bet where With probability, 8/9, you lose , 1, dollars . With probability, 1/9, you win , 9, dollars . The expectation , 1/9, is positive, so conventional wisdom tells you that i\ t is a good deal, and you should accept it, but beware, the standard-deviation is, 3.142696804, that should raise a red flag. Indeed, it you are only allowed to play once, then the probability that you \ would lose money is, 8/9 It would be stupid to take this bet. However, by the law of large numbers, and more quantitavely by the Central L\ imit Theorem, if you are allowed to play many times, you can make the p\ robability of losing money as small as you wish (of course, every gamble carries some rish\ ), the question is how many times should you insist on being able to pla\ y? Let p(n) be the probability that if you play n times, you would gain (at lea\ st some) money Thanks to the amazing Almkvist-Zeilberger algorithm, there is a fast way to \ compute this sequence, via a certain recurrence that we do not display of order, 11, and degree of coefficients , 54 Using this recurrence we can compute the following probabilities If you play, 100, times then your probability of winning a positive amount of money is, 0.5611106948 If you play, 200, times then your probability of winning a positive amount of money is, 0.6413876934 If you play, 300, times then your probability of winning a positive amount of money is, 0.6924632534 If you play, 400, times then your probability of winning a positive amount of money is, 0.7306911832 If you play, 500, times then your probability of winning a positive amount of money is, 0.7612996659 If you play, 600, times then your probability of winning a positive amount of money is, 0.7867253033 If you play, 700, times then your probability of winning a positive amount of money is, 0.8083382016 If you play, 800, times then your probability of winning a positive amount of money is, 0.8270020156 If you play, 900, times then your probability of winning a positive amount of money is, 0.8433045970 If you play, 1000, times then your probability of winning a positive amount of money is, 0.8576687092 If you play, 1100, times then your probability of winning a positive amount of money is, 0.8704111911 If you play, 1200, times then your probability of winning a positive amount of money is, 0.8817771165 If you play, 1300, times then your probability of winning a positive amount of money is, 0.8919607938 If you play, 1400, times then your probability of winning a positive amount of money is, 0.9011195011 If you play, 1500, times then your probability of winning a positive amount of money is, 0.9093826434 If you play, 1600, times then your probability of winning a positive amount of money is, 0.9168582642 If you play, 1700, times then your probability of winning a positive amount of money is, 0.9236374535 If you play, 1800, times then your probability of winning a positive amount of money is, 0.9297978189 If you play, 1900, times then your probability of winning a positive amount of money is, 0.9354061156 If you play, 2000, times then your probability of winning a positive amount of money is, 0.9405202157 ------------------------------- Here is a plot of the number of rounds vs. the probability of winding up wit\ h a positive amount of money + HHHHHHHHHHHHHHHHH + HHHHHHHHHHHHHHHHHHHHHHHHH + HHHHHHHHHHHHHHHHHHH + HHHHHHHHHHHHHHH 0.8+ HHHHHHHHHHHHH + HHHHHHHHHHH + HHHHHHHHHHH +HHHHHHHHHH 0.6+HHHHHHH +HHHHH +HHHH +HHH +HH 0.4+H +H +H +H +H 0.2* * -*---+---+--+---+---+--+---+---+--+---+---+--+---+---+--+---+---+--+---+---+- 0 500 1000 1500 2000 ------------------------------ This ends this chapter that took, 35.434, seconds to generate. Suppose you are offered a bet where With probability, 9/10, you lose , 1, dollars . With probability, 1/10, you win , 10, dollars . The expectation , 1/10, is positive, so conventional wisdom tells you that \ it is a good deal, and you should accept it, but beware, the standard-deviation is, 3.300000000, that should raise a red flag. Indeed, it you are only allowed to play once, then the probability that you \ would lose money is, 9/10 It would be stupid to take this bet. However, by the law of large numbers, and more quantitavely by the Central L\ imit Theorem, if you are allowed to play many times, you can make the p\ robability of losing money as small as you wish (of course, every gamble carries some rish\ ), the question is how many times should you insist on being able to pla\ y? Let p(n) be the probability that if you play n times, you would gain (at lea\ st some) money Thanks to the amazing Almkvist-Zeilberger algorithm, there is a fast way to \ compute this sequence, via a certain recurrence that we do not display of order, 12, and degree of coefficients , 65 Using this recurrence we can compute the following probabilities If you play, 100, times then your probability of winning a positive amount of money is, 0.5487098342 If you play, 200, times then your probability of winning a positive amount of money is, 0.6275804949 If you play, 300, times then your probability of winning a positive amount of money is, 0.6776148576 If you play, 400, times then your probability of winning a positive amount of money is, 0.7151412594 If you play, 500, times then your probability of winning a positive amount of money is, 0.7453105459 If you play, 600, times then your probability of winning a positive amount of money is, 0.7704987935 If you play, 700, times then your probability of winning a positive amount of money is, 0.7920312972 If you play, 800, times then your probability of winning a positive amount of money is, 0.8107378932 If you play, 900, times then your probability of winning a positive amount of money is, 0.8271804988 If you play, 1000, times then your probability of winning a positive amount of money is, 0.8417615486 If you play, 1100, times then your probability of winning a positive amount of money is, 0.8299099704 If you play, 1200, times then your probability of winning a positive amount of money is, 0.8439653958 If you play, 1300, times then your probability of winning a positive amount of money is, 0.8565824076 If you play, 1400, times then your probability of winning a positive amount of money is, 0.8679588561 If you play, 1500, times then your probability of winning a positive amount of money is, 0.8782553456 If you play, 1600, times then your probability of winning a positive amount of money is, 0.8876043262 If you play, 1700, times then your probability of winning a positive amount of money is, 0.8961164063 If you play, 1800, times then your probability of winning a positive amount of money is, 0.9038851802 If you play, 1900, times then your probability of winning a positive amount of money is, 0.9109905690 If you play, 2000, times then your probability of winning a positive amount of money is, 0.9175014160 ------------------------------- Here is a plot of the number of rounds vs. the probability of winding up wit\ h a positive amount of money + HHHHHHHHHHHHHHH + HHHHHHHHHHHHHHHHHHHHHHHHHH + HHHHHHHHHHHHHHHHHHHHH 0.8+ HHHHHHHHHHHHHHHHH + HHHHHHHHHHHHHHH + HHHHHHHHHHHHH + HHHHHHHHHHHH +HHHHHHHHHH 0.6+HHHHHHHH +HHHHH +HHHH +HH +HH 0.4+H +H +H +H +H 0.2* * -*---+---+--+---+---+--+---+---+--+---+---+--+---+---+--+---+---+--+---+---+- 0 500 1000 1500 2000 ------------------------------ This ends this chapter that took, 90.653, seconds to generate. Suppose you are offered a bet where 10 With probability, --, you lose , 1, dollars . 11 With probability, 1/11, you win , 11, dollars . The expectation , 1/11, is positive, so conventional wisdom tells you that \ it is a good deal, and you should accept it, but beware, the standard-deviation is, 3.449757448, that should raise a red flag. Indeed, it you are only allowed to play once, then the probability that you \ 10 would lose money is, -- 11 It would be stupid to take this bet. However, by the law of large numbers, and more quantitavely by the Central L\ imit Theorem, if you are allowed to play many times, you can make the p\ robability of losing money as small as you wish (of course, every gamble carries some rish\ ), the question is how many times should you insist on being able to pla\ y? Let p(n) be the probability that if you play n times, you would gain (at lea\ st some) money Thanks to the amazing Almkvist-Zeilberger algorithm, there is a fast way to \ compute this sequence, via a certain recurrence that we do not display of order, 13, and degree of coefficients , 77 Using this recurrence we can compute the following probabilities If you play, 100, times then your probability of winning a positive amount of money is, 0.5630901565 If you play, 200, times then your probability of winning a positive amount of money is, 0.6499489054 If you play, 300, times then your probability of winning a positive amount of money is, 0.6299381498 If you play, 400, times then your probability of winning a positive amount of money is, 0.6843663097 If you play, 500, times then your probability of winning a positive amount of money is, 0.7263639117 If you play, 600, times then your probability of winning a positive amount of money is, 0.7127295537 If you play, 700, times then your probability of winning a positive amount of money is, 0.7471348827 If you play, 800, times then your probability of winning a positive amount of money is, 0.7760618674 If you play, 900, times then your probability of winning a positive amount of money is, 0.7658588382 If you play, 1000, times then your probability of winning a positive amount of money is, 0.7910430734 If you play, 1100, times then your probability of winning a positive amount of money is, 0.8129202429 If you play, 1200, times then your probability of winning a positive amount of money is, 0.8049054195 If you play, 1300, times then your probability of winning a positive amount of money is, 0.8244784890 If you play, 1400, times then your probability of winning a positive amount of money is, 0.8417773877 If you play, 1500, times then your probability of winning a positive amount of money is, 0.8353002914 If you play, 1600, times then your probability of winning a positive amount of money is, 0.8510188687 If you play, 1700, times then your probability of winning a positive amount of money is, 0.8650602731 If you play, 1800, times then your probability of winning a positive amount of money is, 0.8597279042 If you play, 1900, times then your probability of winning a positive amount of money is, 0.8726146871 If you play, 2000, times then your probability of winning a positive amount of money is, 0.8842103404 ------------------------------- Here is a plot of the number of rounds vs. the probability of winding up wit\ h a positive amount of money + HHHHHHHHHHHHHH + HHHHHHHHHHHHHHHHHHHHHHHHHHH 0.8+ HHHHHHHHHHHHHHHHHHHHHHH + HHHHHHHHHHHHHHHHHHH + HHHHHHHHHHHHHHHHH + HHHHHHHHHHHHHHH +HHHHHHHHHHHHHHH 0.6+HHHHHHHHHHH +HHHHHHHH +HHHHH +HHHH +HHH 0.4+HH +H +H +H +H 0.2+H * * -*---+---+--+---+---+--+---+---+--+---+---+--+---+---+--+---+---+--+---+---+- 0 500 1000 1500 2000 ------------------------------ This ends this chapter that took, 93.770, seconds to generate. Suppose you are offered a bet where 11 With probability, --, you lose , 1, dollars . 12 With probability, 1/12, you win , 12, dollars . The expectation , 1/12, is positive, so conventional wisdom tells you that \ it is a good deal, and you should accept it, but beware, the standard-deviation is, 3.593010188, that should raise a red flag. Indeed, it you are only allowed to play once, then the probability that you \ 11 would lose money is, -- 12 It would be stupid to take this bet. However, by the law of large numbers, and more quantitavely by the Central L\ imit Theorem, if you are allowed to play many times, you can make the p\ robability of losing money as small as you wish (of course, every gamble carries some rish\ ), the question is how many times should you insist on being able to pla\ y? Let p(n) be the probability that if you play n times, you would gain (at lea\ st some) money Thanks to the amazing Almkvist-Zeilberger algorithm, there is a fast way to \ compute this sequence, via a certain recurrence that we do not display of order, 14, and degree of coefficients , 90 Using this recurrence we can compute the following probabilities If you play, 100, times then your probability of winning a positive amount of money is, 0.6002971740 If you play, 200, times then your probability of winning a positive amount of money is, 0.6046494562 If you play, 300, times then your probability of winning a positive amount of money is, 0.6128413305 If you play, 400, times then your probability of winning a positive amount of money is, 0.6892563543 If you play, 500, times then your probability of winning a positive amount of money is, 0.6899075575 If you play, 600, times then your probability of winning a positive amount of money is, 0.6920847098 If you play, 700, times then your probability of winning a positive amount of money is, 0.7422217708 If you play, 800, times then your probability of winning a positive amount of money is, 0.7424320854 If you play, 900, times then your probability of winning a positive amount of money is, 0.7434117521 If you play, 1000, times then your probability of winning a positive amount of money is, 0.7810430864 If you play, 1100, times then your probability of winning a positive amount of money is, 0.7811202697 If you play, 1200, times then your probability of winning a positive amount of money is, 0.7816555570 If you play, 1300, times then your probability of winning a positive amount of money is, 0.7825378441 If you play, 1400, times then your probability of winning a positive amount of money is, 0.8115609317 If you play, 1500, times then your probability of winning a positive amount of money is, 0.8118842997 If you play, 1600, times then your probability of winning a positive amount of money is, 0.8124488317 If you play, 1700, times then your probability of winning a positive amount of money is, 0.8363634820 If you play, 1800, times then your probability of winning a positive amount of money is, 0.8365708797 If you play, 1900, times then your probability of winning a positive amount of money is, 0.8369526225 If you play, 2000, times then your probability of winning a positive amount of money is, 0.8570154002 ------------------------------- Here is a plot of the number of rounds vs. the probability of winding up wit\ h a positive amount of money + HHHHHHHHHHHHHH 0.8+ HHHHHHHHHHHHHHHHHHHHHHHHHHHH + HHHHHHHHHHHHHHHHHHHHHHHHH | HHHHHHHHHHHHHHHHHHHHH + HHHHHHHHHHHHHHHHHHH +H HHHHHHHHHHHHHHHHH +HHHHHHHHHHHHHHHH 0.6+HHHHHHHHHHH +HHHHHHHH +HHHHH +HHH +HH 0.4+HH +H +H +H +H 0.2+H * * -*---+---+--+---+---+--+---+---+--+---+---+--+---+---+--+---+---+--+---+---+- 0 500 1000 1500 2000 ------------------------------ This ends this chapter that took, 343.859, seconds to generate. Suppose you are offered a bet where 12 With probability, --, you lose , 1, dollars . 13 With probability, 1/13, you win , 13, dollars . The expectation , 1/13, is positive, so conventional wisdom tells you that \ it is a good deal, and you should accept it, but beware, the standard-deviation is, 3.730570971, that should raise a red flag. Indeed, it you are only allowed to play once, then the probability that you \ 12 would lose money is, -- 13 It would be stupid to take this bet. However, by the law of large numbers, and more quantitavely by the Central L\ imit Theorem, if you are allowed to play many times, you can make the p\ robability of losing money as small as you wish (of course, every gamble carries some rish\ ), the question is how many times should you insist on being able to pla\ y? Let p(n) be the probability that if you play n times, you would gain (at lea\ st some) money Thanks to the amazing Almkvist-Zeilberger algorithm, there is a fast way to \ compute this sequence, via a certain recurrence that we do not display of order, 15, and degree of coefficients , 104 Using this recurrence we can compute the following probabilities If you play, 100, times then your probability of winning a positive amount of money is, 0.5073319701 If you play, 200, times then your probability of winning a positive amount of money is, 0.5787389508 If you play, 300, times then your probability of winning a positive amount of money is, 0.6233069221 If you play, 400, times then your probability of winning a positive amount of money is, 0.6568091521 If you play, 500, times then your probability of winning a positive amount of money is, 0.6840139911 If you play, 600, times then your probability of winning a positive amount of money is, 0.7070476861 If you play, 700, times then your probability of winning a positive amount of money is, 0.6768636508 If you play, 800, times then your probability of winning a positive amount of money is, 0.6992868172 If you play, 900, times then your probability of winning a positive amount of money is, 0.7191190622 If you play, 1000, times then your probability of winning a positive amount of money is, 0.7368666303 If you play, 1100, times then your probability of winning a positive amount of money is, 0.7528938931 If you play, 1200, times then your probability of winning a positive amount of money is, 0.7674729753 If you play, 1300, times then your probability of winning a positive amount of money is, 0.7808132214 If you play, 1400, times then your probability of winning a positive amount of money is, 0.7625686790 If you play, 1500, times then your probability of winning a positive amount of money is, 0.7758745169 If you play, 1600, times then your probability of winning a positive amount of money is, 0.7881672924 If you play, 1700, times then your probability of winning a positive amount of money is, 0.7995613366 If you play, 1800, times then your probability of winning a positive amount of money is, 0.8101524841 If you play, 1900, times then your probability of winning a positive amount of money is, 0.8200219232 If you play, 2000, times then your probability of winning a positive amount of money is, 0.8292390937 ------------------------------- Here is a plot of the number of rounds vs. the probability of winding up wit\ h a positive amount of money + HHHHHHHHHHHHHH 0.8+ HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH + HHHHHHHHHHHHHHHHHHHHHHHHHH 0.7+ HHHHHHHHHHHHHHHHHHHHHHH + HHHHHHHHHHHHHHHHHHHHH +H HHHHHHHHHHHHHHHHHHH 0.6+HHHHHHHHHHHHHHHH +HHHHHHHHHHH +HHHHHHHH 0.5+HHHHH +HHH 0.4+HH +H +H 0.3+H +H +H 0.2+H *H 0.1* -*---+---+--+---+---+--+---+---+--+---+---+--+---+---+--+---+---+--+---+---+- 0 500 1000 1500 2000 ------------------------------ This ends this chapter that took, 388.513, seconds to generate. Suppose you are offered a bet where 13 With probability, --, you lose , 1, dollars . 14 With probability, 1/14, you win , 14, dollars . The expectation , 1/14, is positive, so conventional wisdom tells you that \ it is a good deal, and you should accept it, but beware, the standard-deviation is, 3.863090650, that should raise a red flag. Indeed, it you are only allowed to play once, then the probability that you \ 13 would lose money is, -- 14 It would be stupid to take this bet. However, by the law of large numbers, and more quantitavely by the Central L\ imit Theorem, if you are allowed to play many times, you can make the p\ robability of losing money as small as you wish (of course, every gamble carries some rish\ ), the question is how many times should you insist on being able to pla\ y? Let p(n) be the probability that if you play n times, you would gain (at lea\ st some) money Thanks to the amazing Almkvist-Zeilberger algorithm, there is a fast way to \ compute this sequence, via a certain recurrence that we do not display of order, 16, and degree of coefficients , 119 Using this recurrence we can compute the following probabilities If you play, 100, times then your probability of winning a positive amount of money is, 0.5775575933 If you play, 200, times then your probability of winning a positive amount of money is, 0.5704714453 If you play, 300, times then your probability of winning a positive amount of money is, 0.5702775137 If you play, 400, times then your probability of winning a positive amount of money is, 0.6474751642 If you play, 500, times then your probability of winning a positive amount of money is, 0.6417218314 If you play, 600, times then your probability of winning a positive amount of money is, 0.6383017478 If you play, 700, times then your probability of winning a positive amount of money is, 0.6907733710 If you play, 800, times then your probability of winning a positive amount of money is, 0.6862431753 If you play, 900, times then your probability of winning a positive amount of money is, 0.6828888301 If you play, 1000, times then your probability of winning a positive amount of money is, 0.7237346743 If you play, 1100, times then your probability of winning a positive amount of money is, 0.7199946322 If you play, 1200, times then your probability of winning a positive amount of money is, 0.7169659220 If you play, 1300, times then your probability of winning a positive amount of money is, 0.7506150395 If you play, 1400, times then your probability of winning a positive amount of money is, 0.7474344791 If you play, 1500, times then your probability of winning a positive amount of money is, 0.7447294731 If you play, 1600, times then your probability of winning a positive amount of money is, 0.7733323096 If you play, 1700, times then your probability of winning a positive amount of money is, 0.7705743784 If you play, 1800, times then your probability of winning a positive amount of money is, 0.7681550227 If you play, 1900, times then your probability of winning a positive amount of money is, 0.7929579188 If you play, 2000, times then your probability of winning a positive amount of money is, 0.7905335552 ------------------------------- Here is a plot of the number of rounds vs. the probability of winding up wit\ h a positive amount of money 0.8+ HHHHHHHHHHHHH + HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH + HHHHHHHHHHHHHHHHHHHHHHHHHHHHH 0.7+ HHHHHHHHHHHHHHHHHHHHHHHHHH +H HHHHHHHHHHHHHHHHHHHHHHH +HHHHHHHHHHHHHHHHHHHHHH 0.6+HHHHHHHHHHHHHHHH +HHHHHHHHHHH 0.5+HHHHHHH +HHHHH +HHH 0.4+HH +HH +H 0.3+H +H +H 0.2+H * 0.1* -*---+---+--+---+---+--+---+---+--+---+---+--+---+---+--+---+---+--+---+---+- 0 500 1000 1500 2000 ------------------------------ This ends this chapter that took, 6499.534, seconds to generate. Suppose you are offered a bet where 14 With probability, --, you lose , 1, dollars . 15 With probability, 1/15, you win , 15, dollars . The expectation , 1/15, is positive, so conventional wisdom tells you that \ it is a good deal, and you should accept it, but beware, the standard-deviation is, 3.991101214, that should raise a red flag. Indeed, it you are only allowed to play once, then the probability that you \ 14 would lose money is, -- 15 It would be stupid to take this bet. However, by the law of large numbers, and more quantitavely by the Central L\ imit Theorem, if you are allowed to play many times, you can make the p\ robability of losing money as small as you wish (of course, every gamble carries some rish\ ), the question is how many times should you insist on being able to pla\ y? Let p(n) be the probability that if you play n times, you would gain (at lea\ st some) money Thanks to the amazing Almkvist-Zeilberger algorithm, there is a fast way to \ compute this sequence, via a certain recurrence that we do not display of order, 17, and degree of coefficients , 135 Using this recurrence we can compute the following probabilities If you play, 100, times then your probability of winning a positive amount of money is, 0.5031085223 If you play, 200, times then your probability of winning a positive amount of money is, 0.5779721083 If you play, 300, times then your probability of winning a positive amount of money is, 0.6244479265 If you play, 400, times then your probability of winning a positive amount of money is, 0.5816519385 If you play, 500, times then your probability of winning a positive amount of money is, 0.6199200174 If you play, 600, times then your probability of winning a positive amount of money is, 0.6514402657 If you play, 700, times then your probability of winning a positive amount of money is, 0.6782342440 If you play, 800, times then your probability of winning a positive amount of money is, 0.6495948864 If you play, 900, times then your probability of winning a positive amount of money is, 0.6745347146 If you play, 1000, times then your probability of winning a positive amount of money is, 0.6966871763 If you play, 1100, times then your probability of winning a positive amount of money is, 0.7165538870 If you play, 1200, times then your probability of winning a positive amount of money is, 0.6944450648 If you play, 1300, times then your probability of winning a positive amount of money is, 0.7134981234 If you play, 1400, times then your probability of winning a positive amount of money is, 0.7308951643 If you play, 1500, times then your probability of winning a positive amount of money is, 0.7468509782 If you play, 1600, times then your probability of winning a positive amount of money is, 0.7287553067 If you play, 1700, times then your probability of winning a positive amount of money is, 0.7442635723 If you play, 1800, times then your probability of winning a positive amount of money is, 0.7586279421 If you play, 1900, times then your probability of winning a positive amount of money is, 0.7719659284 If you play, 2000, times then your probability of winning a positive amount of money is, 0.7566718303 ------------------------------- Here is a plot of the number of rounds vs. the probability of winding up wit\ h a positive amount of money + HHHHHHHHHHHHH + HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH 0.7+ HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH + HHHHHHHHHHHHHHHHHHHHHHHHHHHH +HH HHHHHHHHHHHHHHHHHHHHHHHHHH 0.6+HHHHHHHHHHHHHHHHHHHHHH +HHHHHHHHHHHHHHH +HHHHHHHHHH 0.5+HHHHHHH +HHHHH +HHH 0.4+HH +HH +H 0.3+H +H 0.2+H +H * 0.1* -*---+---+--+---+---+--+---+---+--+---+---+--+---+---+--+---+---+--+---+---+- 0 500 1000 1500 2000 ------------------------------ This ends this chapter that took, 7371.063, seconds to generate. -------------------------------------- This ends this paper that took, 14867.312, seconds to produce.