How many times should a risk-averse gambler inisist on playing with Probabi\ lity table [[-1,(i-1)/i],[i,1/i]] for i from 2 to 10 with risk-aversenes\ s 10^(-j) for j from 1 to , 4 By Shalosh B. Ekhad The above values are based on the Central Limit theorem approximations, and \ are slightlt higher than necessary for the stated risk-averseness ------------------------------------ Suppose you are offered a bet such that You lose, 2, dollar with probability, 1/2, and you win, 4, dollars with probability, 1/2 The expected gain is one dollar, so if you are rational you should accept th\ is bet. Alas you hate to lose money. If you are willing to take a chance of , 1/10, of losing some money then you should insist on playing this game, 15, times If you are willing to take a chance of , 1/100, of losing some money then you should insist on playing this game, 49, times If you are willing to take a chance of , 1/1000, of losing some money then you should insist on playing this game, 86, times If you are willing to take a chance of , 1/10000, of losing some money then you should insist on playing this game, 125, times ------------------------------------ Suppose you are offered a bet such that You lose, 3, dollar with probability, 2/3, and you win, 9, dollars with probability, 1/3 The expected gain is one dollar, so if you are rational you should accept th\ is bet. Alas you hate to lose money. If you are willing to take a chance of , 1/10, of losing some money then you should insist on playing this game, 53, times If you are willing to take a chance of , 1/100, of losing some money then you should insist on playing this game, 174, times If you are willing to take a chance of , 1/1000, of losing some money then you should insist on playing this game, 306, times If you are willing to take a chance of , 1/10000, of losing some money then you should insist on playing this game, 443, times ------------------------------------ Suppose you are offered a bet such that You lose, 4, dollar with probability, 3/4, and you win, 16, dollars with probability, 1/4 The expected gain is one dollar, so if you are rational you should accept th\ is bet. Alas you hate to lose money. If you are willing to take a chance of , 1/10, of losing some money then you should insist on playing this game, 124, times If you are willing to take a chance of , 1/100, of losing some money then you should insist on playing this game, 406, times If you are willing to take a chance of , 1/1000, of losing some money then you should insist on playing this game, 717, times If you are willing to take a chance of , 1/10000, of losing some money then you should insist on playing this game, 1038, times ------------------------------------ Suppose you are offered a bet such that You lose, 5, dollar with probability, 4/5, and you win, 25, dollars with probability, 1/5 The expected gain is one dollar, so if you are rational you should accept th\ is bet. Alas you hate to lose money. If you are willing to take a chance of , 1/10, of losing some money then you should insist on playing this game, 237, times If you are willing to take a chance of , 1/100, of losing some money then you should insist on playing this game, 780, times If you are willing to take a chance of , 1/1000, of losing some money then you should insist on playing this game, 1376, times If you are willing to take a chance of , 1/10000, of losing some money then you should insist on playing this game, 1992, times ------------------------------------ Suppose you are offered a bet such that You lose, 6, dollar with probability, 5/6, and you win, 36, dollars with probability, 1/6 The expected gain is one dollar, so if you are rational you should accept th\ is bet. Alas you hate to lose money. If you are willing to take a chance of , 1/10, of losing some money then you should insist on playing this game, 403, times If you are willing to take a chance of , 1/100, of losing some money then you should insist on playing this game, 1326, times If you are willing to take a chance of , 1/1000, of losing some money then you should insist on playing this game, 2340, times If you are willing to take a chance of , 1/10000, of losing some money then you should insist on playing this game, 3389, times ------------------------------------ Suppose you are offered a bet such that You lose, 7, dollar with probability, 6/7, and you win, 49, dollars with probability, 1/7 The expected gain is one dollar, so if you are rational you should accept th\ is bet. Alas you hate to lose money. If you are willing to take a chance of , 1/10, of losing some money then you should insist on playing this game, 631, times If you are willing to take a chance of , 1/100, of losing some money then you should insist on playing this game, 2079, times If you are willing to take a chance of , 1/1000, of losing some money then you should insist on playing this game, 3668, times If you are willing to take a chance of , 1/10000, of losing some money then you should insist on playing this game, 5312, times ------------------------------------ Suppose you are offered a bet such that You lose, 8, dollar with probability, 7/8, and you win, 64, dollars with probability, 1/8 The expected gain is one dollar, so if you are rational you should accept th\ is bet. Alas you hate to lose money. If you are willing to take a chance of , 1/10, of losing some money then you should insist on playing this game, 932, times If you are willing to take a chance of , 1/100, of losing some money then you should insist on playing this game, 3069, times If you are willing to take a chance of , 1/1000, of losing some money then you should insist on playing this game, 5415, times If you are willing to take a chance of , 1/10000, of losing some money then you should insist on playing this game, 7843, times ------------------------------------ Suppose you are offered a bet such that You lose, 9, dollar with probability, 8/9, and you win, 81, dollars with probability, 1/9 The expected gain is one dollar, so if you are rational you should accept th\ is bet. Alas you hate to lose money. If you are willing to take a chance of , 1/10, of losing some money then you should insist on playing this game, 1314, times If you are willing to take a chance of , 1/100, of losing some money then you should insist on playing this game, 4330, times If you are willing to take a chance of , 1/1000, of losing some money then you should insist on playing this game, 7640, times If you are willing to take a chance of , 1/10000, of losing some money then you should insist on playing this game, 11065, times ------------------------------------ Suppose you are offered a bet such that You lose, 10, dollar with probability, 9/10, and you win, 100, dollars with probability, 1/10 The expected gain is one dollar, so if you are rational you should accept th\ is bet. Alas you hate to lose money. If you are willing to take a chance of , 1/10, of losing some money then you should insist on playing this game, 1789, times If you are willing to take a chance of , 1/100, of losing some money then you should insist on playing this game, 5894, times If you are willing to take a chance of , 1/1000, of losing some money then you should insist on playing this game, 10400, times If you are willing to take a chance of , 1/10000, of losing some money then you should insist on playing this game, 15063, times ------------------------------------ Suppose you are offered a bet such that 10 You lose, 11, dollar with probability, --, and you win, 121, 11 dollars with probability, 1/11 The expected gain is one dollar, so if you are rational you should accept th\ is bet. Alas you hate to lose money. If you are willing to take a chance of , 1/10, of losing some money then you should insist on playing this game, 2366, times If you are willing to take a chance of , 1/100, of losing some money then you should insist on playing this game, 7794, times If you are willing to take a chance of , 1/1000, of losing some money then you should insist on playing this game, 13752, times If you are willing to take a chance of , 1/10000, of losing some money then you should insist on playing this game, 19917, times ------------------------------------ Suppose you are offered a bet such that 11 You lose, 12, dollar with probability, --, and you win, 144, 12 dollars with probability, 1/12 The expected gain is one dollar, so if you are rational you should accept th\ is bet. Alas you hate to lose money. If you are willing to take a chance of , 1/10, of losing some money then you should insist on playing this game, 3054, times If you are willing to take a chance of , 1/100, of losing some money then you should insist on playing this game, 10061, times If you are willing to take a chance of , 1/1000, of losing some money then you should insist on playing this game, 17753, times If you are willing to take a chance of , 1/10000, of losing some money then you should insist on playing this game, 25712, times ------------------------------------ Suppose you are offered a bet such that 12 You lose, 13, dollar with probability, --, and you win, 169, 13 dollars with probability, 1/13 The expected gain is one dollar, so if you are rational you should accept th\ is bet. Alas you hate to lose money. If you are willing to take a chance of , 1/10, of losing some money then you should insist on playing this game, 3863, times If you are willing to take a chance of , 1/100, of losing some money then you should insist on playing this game, 12729, times If you are willing to take a chance of , 1/1000, of losing some money then you should insist on playing this game, 22461, times If you are willing to take a chance of , 1/10000, of losing some money then you should insist on playing this game, 32531, times ------------------------------------ Suppose you are offered a bet such that 13 You lose, 14, dollar with probability, --, and you win, 196, 14 dollars with probability, 1/14 The expected gain is one dollar, so if you are rational you should accept th\ is bet. Alas you hate to lose money. If you are willing to take a chance of , 1/10, of losing some money then you should insist on playing this game, 4804, times If you are willing to take a chance of , 1/100, of losing some money then you should insist on playing this game, 15830, times If you are willing to take a chance of , 1/1000, of losing some money then you should insist on playing this game, 27933, times If you are willing to take a chance of , 1/10000, of losing some money then you should insist on playing this game, 40456, times ------------------------------------ Suppose you are offered a bet such that 14 You lose, 15, dollar with probability, --, and you win, 225, 15 dollars with probability, 1/15 The expected gain is one dollar, so if you are rational you should accept th\ is bet. Alas you hate to lose money. If you are willing to take a chance of , 1/10, of losing some money then you should insist on playing this game, 5887, times If you are willing to take a chance of , 1/100, of losing some money then you should insist on playing this game, 19397, times If you are willing to take a chance of , 1/1000, of losing some money then you should insist on playing this game, 34226, times If you are willing to take a chance of , 1/10000, of losing some money then you should insist on playing this game, 49571, times ------------------------------------ Suppose you are offered a bet such that 15 You lose, 16, dollar with probability, --, and you win, 256, 16 dollars with probability, 1/16 The expected gain is one dollar, so if you are rational you should accept th\ is bet. Alas you hate to lose money. If you are willing to take a chance of , 1/10, of losing some money then you should insist on playing this game, 7120, times If you are willing to take a chance of , 1/100, of losing some money then you should insist on playing this game, 23461, times If you are willing to take a chance of , 1/1000, of losing some money then you should insist on playing this game, 41398, times If you are willing to take a chance of , 1/10000, of losing some money then you should insist on playing this game, 59958, times ------------------------------------ Suppose you are offered a bet such that 16 You lose, 17, dollar with probability, --, and you win, 289, 17 dollars with probability, 1/17 The expected gain is one dollar, so if you are rational you should accept th\ is bet. Alas you hate to lose money. If you are willing to take a chance of , 1/10, of losing some money then you should insist on playing this game, 8515, times If you are willing to take a chance of , 1/100, of losing some money then you should insist on playing this game, 28056, times If you are willing to take a chance of , 1/1000, of losing some money then you should insist on playing this game, 49505, times If you are willing to take a chance of , 1/10000, of losing some money then you should insist on playing this game, 71701, times ------------------------------------ Suppose you are offered a bet such that 17 You lose, 18, dollar with probability, --, and you win, 324, 18 dollars with probability, 1/18 The expected gain is one dollar, so if you are rational you should accept th\ is bet. Alas you hate to lose money. If you are willing to take a chance of , 1/10, of losing some money then you should insist on playing this game, 10080, times If you are willing to take a chance of , 1/100, of losing some money then you should insist on playing this game, 33213, times If you are willing to take a chance of , 1/1000, of losing some money then you should insist on playing this game, 58606, times If you are willing to take a chance of , 1/10000, of losing some money then you should insist on playing this game, 84882, times ------------------------------------ Suppose you are offered a bet such that 18 You lose, 19, dollar with probability, --, and you win, 361, 19 dollars with probability, 1/19 The expected gain is one dollar, so if you are rational you should accept th\ is bet. Alas you hate to lose money. If you are willing to take a chance of , 1/10, of losing some money then you should insist on playing this game, 11826, times If you are willing to take a chance of , 1/100, of losing some money then you should insist on playing this game, 38966, times If you are willing to take a chance of , 1/1000, of losing some money then you should insist on playing this game, 68757, times If you are willing to take a chance of , 1/10000, of losing some money then you should insist on playing this game, 99584, times ------------------------------------ Suppose you are offered a bet such that 19 You lose, 20, dollar with probability, --, and you win, 400, 20 dollars with probability, 1/20 The expected gain is one dollar, so if you are rational you should accept th\ is bet. Alas you hate to lose money. If you are willing to take a chance of , 1/10, of losing some money then you should insist on playing this game, 13762, times If you are willing to take a chance of , 1/100, of losing some money then you should insist on playing this game, 45347, times If you are willing to take a chance of , 1/1000, of losing some money then you should insist on playing this game, 80016, times If you are willing to take a chance of , 1/10000, of losing some money then you should insist on playing this game, 115891, times -------------------------------------- This ends this paper that took, 2.223, seconds to produce. ----------------------------- This took, 2.355, seconds.