# 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 rounded to 7 decimal places behind
the decimal point in the form 0.abcdefg.