Problem 499
St. Petersburg Lottery
A gambler decides to participate in a special lottery. In this lottery the gambler plays a series of one or more games.
Each game costs m pounds to play and starts with an initial pot of 1 pound. The gambler flips an unbiased coin. Every time a head appears, the pot is doubled and the gambler continues. When a tail appears, the game ends and the gambler collects the current value of the pot. The gambler is certain to win at least 1 pound, the starting value of the pot, at the cost of m pounds, the initial fee.
The gambler cannot continue to play if his fortune falls below m pounds.
Let pm(s) denote the probability that the gambler will never run out of money in this lottery given his initial fortune s and the cost per game m.
For example p2(2) = 0.2522, p2(5) = 0.6873 and p6(10 000) = 0.9952 (note: pm(s) = 0 for s < m).
Find p15(109) and give your answer in the form 0.abcdefg
圣彼得堡彩票
一名赌徒决定去买一种特殊的彩票。这种彩票要求赌徒玩一系列的游戏。
每次进行游戏都会花去赌徒m英镑。游戏开始时，赌徒将持有1英镑，然后他将掷出一枚公平的硬币：如果掷出字，持有的钱翻倍，游戏继续；如果掷出花，游戏结束，赌徒得到自己所持有的钱数。也就是说，赌徒每次游戏将花费m英镑，但将确保至少能够得到1英镑。
如果赌徒的财产已经不足m英镑了，他将不能继续进行游戏。
当赌徒初始的财产为s，并且每次游戏的花费是m时，记pm(s)为赌徒永远不会不够钱继续游戏的概率。
例如，p2(2) = 0.2522，p2(5) = 0.6873，p6(10 000) = 0.9952（注意：如果s < m，则pm(s) = 0）。
求p15(109)，用0.abcdefg的形式给出你的答案。