Problem 232
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 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.
赛跑游戏
两名玩家轮流抛掷一枚标准硬币进行“赛跑游戏”。轮到玩家1时，他掷硬币1次：若硬币是正面朝上则得1分；若硬币是反面朝上则不得分。轮到玩家2时，她指定一个正整数T，并掷硬币T次：若硬币每次均正面朝上，则她得2T-1分；否则她不得分。玩家1先行。首先到达100分或更多的玩家获胜。
每一轮，玩家2总是选择最大化自己获胜概率的掷硬币次数T。
玩家2获胜的概率是多少？
将你的答案四舍五入到八位小数，即格式为0.abcdefgh。