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 game ends if the gambler's fortune falls below $m$ pounds.
Let $p_m(s)$ denote the probability that the gambler will never run out of
money in this lottery given an initial fortune $s$ and the cost per game $m$.
For example $p_2(2) \approx 0.2522$, $p_2(5) \approx 0.6873$ and
$p_6(10,000) \approx 0.9952$ (note: $p_m(s) = 0$ for $s \lt m$).
Find $p_{15}(10^9)$ and give your answer rounded to $7$ decimal places behind
the decimal point in the form 0.abcdefg.