Problem 898

Claire Voyant

Claire Voyant is a teacher playing a game with a class of students. A fair coin is tossed on the table. All the students can see the outcome of the toss, but Claire cannot. Each student then tells Claire whether the outcome is head or tail. The students may lie, but Claire knows the probability that each individual student lies. Moreover, the students lie independently. After that, Claire attempts to guess the outcome using an optimal strategy.

For example, for a class of four students with lying probabilities $20\%,40\%,60\%,80\%$, Claire guesses correctly with probability $0.832$.

Find the probability that Claire guesses correctly for a class of $51$ students each lying with a probability of $25\%, 26\%, \dots, 75\%$ respectively.

Give your answer rounded to $10$ digits after the decimal point.

克莱尔·瓦杨

克莱尔·瓦杨老师正在和全班学生玩一种游戏：抛掷一枚公平硬币，所有学生都能看到投掷的结果，但克莱尔看不到。然后，每个学生依次告诉克莱尔结果是正面还是反面。学生可能会撒谎，但克莱尔知道每个学生撒谎的概率。此外，每个学生是否撒谎是相互独立的。之后，克莱尔尝试用最优策略猜测硬币的正反面。

例如，若全班共有四名学生，他们撒谎的概率分别是$20\%$、$40\%$、$60\%$、$80\%$，克莱尔猜对的概率是$0.832$。

若全班共有$51$名学生，每个学生撒谎的概率分别是$25\%$、$26\%$、$\ldots$、$75\%$，求克莱尔猜对的概率。

将你的答案四舍五入保留$10$位小数。

译注：本题中老师的名字是对“Clairvoyant/先知”一词的文字游戏。