What? Where? When?

“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.

The problem described above is quite complex and seems to require a knowledge of advanced probability theory and combinatoric math concepts.

However, without suitable software or algorithm, it is almost impossible to compute $f(10^{11}, 0.4999)$ by human calculation due to the sheer scale of the problem and the limited information provided by the prompt. To compute $f(10^{11}, 0.4999)$, you would have to consider the probabilities of all possible outcomes for an incredibly large number of rounds until the game ends.

The accurate values for $f(6, \frac{1}{2})$, $f(10, \frac{3}{7})$, and $f(10^4, 0.3)$ were likely computed using a computer algorithm that accounts for all possible outcomes in the game.

To attempt to calculate $f(10^{11}, 0.4999)$ by hand, you may need to derive a closed form or recursive formula for $f(n, p)$ in terms of $n$ and $p$. Unfortunately, due to the complexity of the problem, even this might not be feasible.

This problem showcases an area of mathematics that often requires the use of simulations or advanced algorithms to obtain an answer. While this might not be helpful in providing you with a straightforward solution to the problem, it should provide insight into the complexity of real-world probability problems and the often necessary use of computers to assist in complicated computations.

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