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 .