Advice to the risk averse gambler about playing at the St. Petersburg casino\ with i rounds from i=4 to i=, 9 By Shalosh B. Ekhad ---------------------------------------------------- Suppose you are at St. Petersburg and are ready to enter its famous casino, \ where there are, 4, rounds With prob. 1/2 you win two ducats, and must leave If you still here, With prob. 1/4 you four two ducats, and must leave etc. If you survived, 4, rounds, you get, 16, ducats Note that the expected gain is, 4, ducats, hence if you pay an entrance fee of, 3, ducats you still expect to win one ducat, so if you are rational you s\ hould play, or shold you? Depending on your risk-averseness, you should insist on the permission to pl\ ay it multiple times So here is the probability distribution With probability, 1/2, you lose , 1, dollars . With probability, 1/4, you win , 1, dollars . With probability, 1/8, you win , 5, dollars . With probability, 1/8, you win , 5, dollars . The expectation is indeed 1, so conventional wisdom tells you that it is \ a good deal, and you should accept it, but beware, the standard-deviation is, 2.449489743, 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? If you play, 50, times then your probability of winning a positive amount of money is, 0.9987389321 If you play, 100, times then your probability of winning a positive amount of money is, 0.9999930834 If you play, 150, times then your probability of winning a positive amount of money is, 0.9999999570 If you play, 200, times then your probability of winning a positive amount of money is, 0.9999999997 If you play, 250, times then your probability of winning a positive amount of money is, 1.000000000 If you play, 300, times then your probability of winning a positive amount of money is, 1.000000000 If you play, 350, times then your probability of winning a positive amount of money is, 1.000000000 If you play, 400, times then your probability of winning a positive amount of money is, 1.000000000 If you play, 450, times then your probability of winning a positive amount of money is, 1.000000000 If you play, 500, times then your probability of winning a positive amount of money is, 1.000000000 ------------------------------- Here is a plot of the number of rounds vs. the probability of winding up wit\ h a positive amount of money 1+ HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH + HH + HH + H 0.9+ H +HH +HH +H 0.8+H +H +H +H 0.7+H +H +H +H 0.6+H +H +H +H 0.5**-+--+--+--+--+--+--+--+--+--+--+-+--+--+--+--+--+--+--+--+--+--+--+--+--+- 0 100 200 300 400 500 ------------------------------ This ends this chapter that took, 8.811, seconds to generate. Suppose you are at St. Petersburg and are ready to enter its famous casino, \ where there are, 5, rounds With prob. 1/2 you win two ducats, and must leave If you still here, With prob. 1/4 you four two ducats, and must leave etc. If you survived, 5, rounds, you get, 32, ducats Note that the expected gain is, 5, ducats, hence if you pay an entrance fee of, 4, ducats you still expect to win one ducat, so if you are rational you s\ hould play, or shold you? Depending on your risk-averseness, you should insist on the permission to pl\ ay it multiple times So here is the probability distribution With probability, 1/2, you lose , 2, dollars . With probability, 1/4, you win , 0, dollars . With probability, 1/8, you win , 4, dollars . With probability, 1/16, you win , 12, dollars . With probability, 1/16, you win , 12, dollars . The expectation is indeed 1, so conventional wisdom tells you that it is \ a good deal, and you should accept it, but beware, the standard-deviation is, 4.582575695, 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? If you play, 50, times then your probability of winning a positive amount of money is, 0.9417210139 If you play, 100, times then your probability of winning a positive amount of money is, 0.9887336872 If you play, 150, times then your probability of winning a positive amount of money is, 0.9975931418 If you play, 200, times then your probability of winning a positive amount of money is, 0.9994622210 If you play, 250, times then your probability of winning a positive amount of money is, 0.9998766885 If you play, 300, times then your probability of winning a positive amount of money is, 0.9999712415 If you play, 350, times then your probability of winning a positive amount of money is, 0.9999932121 If you play, 400, times then your probability of winning a positive amount of money is, 0.9999983834 If you play, 450, times then your probability of winning a positive amount of money is, 0.9999996123 If you play, 500, times then your probability of winning a positive amount of money is, 0.9999999065 ------------------------------- Here is a plot of the number of rounds vs. the probability of winding up wit\ h a positive amount of money 1+ HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH + HHHHH + HHH 0.9+ HHH + H 0.8+ H + HH + H 0.7+ H +H +H 0.6+H +H 0.5+H +H +H 0.4+H +H +H 0.3* -*--+--+--+--+--+--+--+--+--+--+--+-+--+--+--+--+--+--+--+--+--+--+--+--+--+- 0 100 200 300 400 500 ------------------------------ This ends this chapter that took, 44.855, seconds to generate. Suppose you are at St. Petersburg and are ready to enter its famous casino, \ where there are, 6, rounds With prob. 1/2 you win two ducats, and must leave If you still here, With prob. 1/4 you four two ducats, and must leave etc. If you survived, 6, rounds, you get, 64, ducats Note that the expected gain is, 6, ducats, hence if you pay an entrance fee of, 5, ducats you still expect to win one ducat, so if you are rational you s\ hould play, or shold you? Depending on your risk-averseness, you should insist on the permission to pl\ ay it multiple times So here is the probability distribution With probability, 1/2, you lose , 3, dollars . With probability, 1/4, you lose , 1, dollars . With probability, 1/8, you win , 3, dollars . With probability, 1/16, you win , 11, dollars . With probability, 1/32, you win , 27, dollars . With probability, 1/32, you win , 27, dollars . The expectation is indeed 1, so conventional wisdom tells you that it is \ a good deal, and you should accept it, but beware, the standard-deviation is, 7.615773106, 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? If you play, 50, times then your probability of winning a positive amount of money is, 0.8148405285 If you play, 100, times then your probability of winning a positive amount of money is, 0.9088286275 If you play, 150, times then your probability of winning a positive amount of money is, 0.9515883011 If you play, 200, times then your probability of winning a positive amount of money is, 0.9733818383 If you play, 250, times then your probability of winning a positive amount of money is, 0.9850596892 If you play, 300, times then your probability of winning a positive amount of money is, 0.9914979442 If you play, 350, times then your probability of winning a positive amount of money is, 0.9951135543 If you play, 400, times then your probability of winning a positive amount of money is, 0.9971704380 If you play, 450, times then your probability of winning a positive amount of money is, 0.9983518331 If you play, 500, times then your probability of winning a positive amount of money is, 0.9990354044 ------------------------------- Here is a plot of the number of rounds vs. the probability of winding up wit\ h a positive amount of money 1+ HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH + HHHHHHHHHHHHH + HHHHHHH 0.9+ HHHH + HHHH 0.8+ HHH + HH + HH 0.7+ HH + HH + H 0.6+ H +HH 0.5+H +H +H 0.4+H +H +H 0.3+H -*--+--+--+--+--+--+--+--+--+--+--+-+--+--+--+--+--+--+--+--+--+--+--+--+--+- 0 100 200 300 400 500 ------------------------------ This ends this chapter that took, 218.418, seconds to generate. Suppose you are at St. Petersburg and are ready to enter its famous casino, \ where there are, 7, rounds With prob. 1/2 you win two ducats, and must leave If you still here, With prob. 1/4 you four two ducats, and must leave etc. If you survived, 7, rounds, you get, 128, ducats Note that the expected gain is, 7, ducats, hence if you pay an entrance fee of, 6, ducats you still expect to win one ducat, so if you are rational you s\ hould play, or shold you? Depending on your risk-averseness, you should insist on the permission to pl\ ay it multiple times So here is the probability distribution With probability, 1/2, you lose , 4, dollars . With probability, 1/4, you lose , 2, dollars . With probability, 1/8, you win , 2, dollars . With probability, 1/16, you win , 10, dollars . With probability, 1/32, you win , 26, dollars . With probability, 1/64, you win , 58, dollars . With probability, 1/64, you win , 58, dollars . The expectation is indeed 1, so conventional wisdom tells you that it is \ a good deal, and you should accept it, but beware, the standard-deviation is, 11.87434209, 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? If you play, 50, times then your probability of winning a positive amount of money is, 0.6969888465 If you play, 100, times then your probability of winning a positive amount of money is, 0.7915263454 If you play, 150, times then your probability of winning a positive amount of money is, 0.8479068601 If you play, 200, times then your probability of winning a positive amount of money is, 0.8860248435 If you play, 250, times then your probability of winning a positive amount of money is, 0.9132163161 If you play, 300, times then your probability of winning a positive amount of money is, 0.9332062035 If you play, 350, times then your probability of winning a positive amount of money is, 0.9481884094 If you play, 400, times then your probability of winning a positive amount of money is, 0.9595695052 If you play, 450, times then your probability of winning a positive amount of money is, 0.9683011831 If you play, 500, times then your probability of winning a positive amount of money is, 0.9750513738 ------------------------------- Here is a plot of the number of rounds vs. the probability of winding up wit\ h a positive amount of money + HHHHHHHHHHHHHHHHHHH + HHHHHHHHHHHHHHHHHH 0.9+ HHHHHHHHHHH + HHHHHHHH + HHHHHH 0.8+ HHHHH + HHHH + HHH 0.7+ HHH + HH 0.6+ HHH + H + HH 0.5+ H +HH +H 0.4+H +H +H 0.3+H -**-+--+--+--+--+--+--+--+--+--+--+-+--+--+--+--+--+--+--+--+--+--+--+--+--+- 0 100 200 300 400 500 ------------------------------ This ends this chapter that took, 1068.296, seconds to generate. Suppose you are at St. Petersburg and are ready to enter its famous casino, \ where there are, 8, rounds With prob. 1/2 you win two ducats, and must leave If you still here, With prob. 1/4 you four two ducats, and must leave etc. If you survived, 8, rounds, you get, 256, ducats Note that the expected gain is, 8, ducats, hence if you pay an entrance fee of, 7, ducats you still expect to win one ducat, so if you are rational you s\ hould play, or shold you? Depending on your risk-averseness, you should insist on the permission to pl\ ay it multiple times So here is the probability distribution With probability, 1/2, you lose , 5, dollars . With probability, 1/4, you lose , 3, dollars . With probability, 1/8, you win , 1, dollars . With probability, 1/16, you win , 9, dollars . With probability, 1/32, you win , 25, dollars . With probability, 1/64, you win , 57, dollars . With probability, 1/128, you win , 121, dollars . With probability, 1/128, you win , 121, dollars . The expectation is indeed 1, so conventional wisdom tells you that it is \ a good deal, and you should accept it, but beware, the standard-deviation is, 17.83255450, 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? If you play, 50, times then your probability of winning a positive amount of money is, 0.6072894444 If you play, 100, times then your probability of winning a positive amount of money is, 0.6864676323 If you play, 150, times then your probability of winning a positive amount of money is, 0.7385488376 If you play, 200, times then your probability of winning a positive amount of money is, 0.7772315750 If you play, 250, times then your probability of winning a positive amount of money is, 0.8077820183 If you play, 300, times then your probability of winning a positive amount of money is, 0.8327359269 If you play, 350, times then your probability of winning a positive amount of money is, 0.8535560738 If you play, 400, times then your probability of winning a positive amount of money is, 0.8711830368 If you play, 450, times then your probability of winning a positive amount of money is, 0.8862666598 If you play, 500, times then your probability of winning a positive amount of money is, 0.8992792852 ------------------------------- Here is a plot of the number of rounds vs. the probability of winding up wit\ h a positive amount of money 0.9+ HHHHHHHHHHH + HHHHHHHHHHHHHHH + HHHHHHHHHHHH 0.8+ HHHHHHHHHH + HHHHHHHH + HHHHHHH 0.7+ HHHHH + HHHH + HHHH 0.6+ HHH + HH + HH 0.5+ H + HH + HH 0.4+ H +HH +H +H 0.3+H -**-+--+--+--+--+--+--+--+--+--+--+-+--+--+--+--+--+--+--+--+--+--+--+--+--+- 0 100 200 300 400 500 ------------------------------ This ends this chapter that took, 5055.875, seconds to generate. Suppose you are at St. Petersburg and are ready to enter its famous casino, \ where there are, 9, rounds With prob. 1/2 you win two ducats, and must leave If you still here, With prob. 1/4 you four two ducats, and must leave etc. If you survived, 9, rounds, you get, 512, ducats Note that the expected gain is, 9, ducats, hence if you pay an entrance fee of, 8, ducats you still expect to win one ducat, so if you are rational you s\ hould play, or shold you? Depending on your risk-averseness, you should insist on the permission to pl\ ay it multiple times So here is the probability distribution With probability, 1/2, you lose , 6, dollars . With probability, 1/4, you lose , 4, dollars . With probability, 1/8, you win , 0, dollars . With probability, 1/16, you win , 8, dollars . With probability, 1/32, you win , 24, dollars . With probability, 1/64, you win , 56, dollars . With probability, 1/128, you win , 120, dollars . With probability, 1/256, you win , 248, dollars . With probability, 1/256, you win , 248, dollars . The expectation is indeed 1, so conventional wisdom tells you that it is \ a good deal, and you should accept it, but beware, the standard-deviation is, 26.17250466, 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? If you play, 50, times then your probability of winning a positive amount of money is, 0.5153648961 If you play, 100, times then your probability of winning a positive amount of money is, 0.6031097892 If you play, 150, times then your probability of winning a positive amount of money is, 0.6462688996 If you play, 200, times then your probability of winning a positive amount of money is, 0.6800930215 If you play, 250, times then your probability of winning a positive amount of money is, 0.7074877736 If you play, 300, times then your probability of winning a positive amount of money is, 0.7306620863 If you play, 350, times then your probability of winning a positive amount of money is, 0.7507596160 If you play, 400, times then your probability of winning a positive amount of money is, 0.7684837240 If you play, 450, times then your probability of winning a positive amount of money is, 0.7843056534 If you play, 500, times then your probability of winning a positive amount of money is, 0.7985592329 ------------------------------- Here is a plot of the number of rounds vs. the probability of winding up wit\ h a positive amount of money 0.8+ HHHHHHHHHHH + HHHHHHHHHHHHHHH + HHHHHHHHHHHH 0.7+ HHHHHHHHHH + HHHHHHHHH + HHHHHHH 0.6+ HHHHH + HHHH + HHH 0.5+ HH + HH + H 0.4+ HH + HH + H 0.3+HH +H +H 0.2+H * -*--+--+--+--+--+--+--+--+--+--+--+-+--+--+--+--+--+--+--+--+--+--+--+--+--+- 0 100 200 300 400 500 ------------------------------ This ends this chapter that took, 25586.506, seconds to generate. -------------------------------------- This ends this paper that took, 31982.761, seconds to produce.