# Problem 232: The Race

![graphic](img232.gif)

Two players share an unbiased coin and take it in turns to play "The
Race". On Player 1's turn, he tosses the coin once: if it comes up
Heads, he scores one point; if it comes up Tails, he scores nothing. On
Player 2's turn, she chooses a positive integer T and tosses the coin T
times: if it comes up all Heads, she scores 2T-1 points; otherwise, she
scores nothing. Player 1 goes first. The winner is the first to 100 or
more points. On each turn Player 2 selects the number, T, of coin tosses
that maximises the probability of her winning. What is the probability
that Player 2 wins? Give your answer rounded to eight decimal places in
the form 0.abcdefgh .