Problem 499

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



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)。