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What? Where? When?

 Published on Sunday, 24th January 2021, 01:00 am; Solved by 229;
Difficulty rating: 35%

Problem 744

"What? Where? When?" is a TV game show in which a team of experts attempt to answer questions. The following is a simplified version of the game.

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.