Compute probability match lasts 5 games

Q: Compute probability match lasts 5 games

This question evaluates understanding of discrete probability, sequential best-of-n match modeling, Monte Carlo simulation for probability estimation, and interval estimation for proportions.

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Question

Two players play independent games of table tennis.

Player A wins each game with probability $p$ (constant across games).
They play a best-of-5 match (first to 3 wins).
Let $X$ be the number of games played until the match ends (so $X\in\{3,4,5\}$ ).

Tasks

Compute $\theta = \Pr(X=5)$ .
Write a simulation (Monte Carlo) approach to estimate $\theta$ .
Follow-up: given $M$ simulated matches, how would you build a confidence interval for $\theta$ ?

Compute probability match lasts 5 games

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