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Problem 232


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