Problem 744

What? Where? When?

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

It begins with $2 n + 1$ envelopes. $2 n$ 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 $2 n + 1$ 个信封，其中 $2 n$ 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。