## St. Petersburg Lottery

### Problem 499

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) ≈ 0.2522,

`p`

_{2}(5) ≈ 0.6873 and

`p`

_{6}(10 000) ≈ 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.