Problem 744

What? Where? When?

“What? Where? When?” is an TV game show in which an expert attempts to answer questions.

It begins with $2n+1$ envelopes. $2n$ of them contain a question and one contains a RED card.

In each round one of the remaining envelopes is randomly chosen. If the envelope contains the RED card the game ends. If the envelope contains a question the expert gives their answer. If their answer is correct they earn one point, otherwise the viewers earn one point. The game ends normally when either the expert obtains $n$ points or the viewers obtain $n$ points.

Assuming that the expert provides the correct answer with a fixed probability $p$, let $f(n,p)$ be the probability that the game ends normally (i.e. RED card never turns up).

You are given (rounded to $10$ decimal places) that
$f(6,\frac{1}{2})=0.2851562500$,
$f(10,\frac{3}{7})=0.2330040743$,
$f(10^4,0.3)=0.2857499982$.

Find $f(10^{11},0.4999)$. Give your answer rounded to $10$ places behind the decimal point.

什么？哪儿？啥时候？

“什么？哪儿？啥时候？”是一款专家与观众对抗的问答类真人秀电视节目。

节目开始时有$2n+1$个信封，其中$2n$个信封中装有问题，剩下的信封中装有一张红牌。

每一轮随机抽取一个未拆封的信封，如果信封里装有红牌，则游戏结束，若装有问题，则专家必须回答这个问题，答对则专家加一分，答错则观众加一分。如果一直没有抽到红牌，则当专家或观众任意一方获得$n$分时，游戏正常结束。

假设专家答对的概率为固定值$p$，并记$f(n,p)$为游戏正常结束的概率（也即游戏结束前未抽到红牌）。

已知（保留小数点后$10$位）
$f(6,\frac{1}{2})=0.2851562500$，
$f(10,\frac{3}{7})=0.2330040743$，
$f(10^4,0.3)=0.2857499982$。

求$f(10^{11},0.4999)$，并保留小数点后$10$位。