题目232：抛硬币比赛

欧拉计划

The Race

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 2^T-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 .

题目：

两个玩家用一个均匀的硬币玩“抛硬币比赛”。在玩家一的回合，他抛一次硬币：如果得到正面，他得一分；如果得到反而，他得零分。在玩家二的回合，她选择一个正整数 T 并抛 T 次硬币：如果全为正面，她得 2^T-1 分；否则，她得零分。玩家一先抛。最先得到 100 或更多分的玩家获胜。

假设在每一回合玩家二都选择最大化获胜概率的正整数 T 。

那么玩家二获胜的概率为多少？

给出你的答案并精确到 8 位小数，形式为 0.abcdefgh