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 $p_6(10000)\approx 0.9952$ (note: $p_m(s)=0$ for $s<m$).

Find $p_{15}({10}^9)$ and give your answer rounded to $7$ decimal places behind the decimal point in the form 0.abcdefg.