This took me longer to set up than it should have.
First, clarify that the match ends when one player wins 3 games, so going to 5 games means the score is 2-2 after 4 games. Then compute the probability of reaching 2-2 and multiply by the probability that the 5th game is played (which is 1, since the match must end).
Pro tip: Emphasize that the 5th game's outcome doesn't affect the probability of the match going to 5 games; it's solely determined by the first 4 games. Also, mention that this is a negative binomial or binomial scenario, showing you can connect probability to real-world experimentation.
The match goes to exactly 5 games if and only if after 4 games, both players have won exactly 2 games each.
The number of games player A wins in the first 4 games follows a Binomial(4, p) distribution, assuming independence.
Calculate P(A wins exactly 2 games) = C(4,2) * p^2 * (1-p)^2 = 6 p^2 (1-p)^2.
Since the match cannot end before 5 games if the score is 2-2, the probability is exactly the probability of a 2-2 score after 4 games.
The probability is 6 p^2 (1-p)^2. Optionally, discuss edge cases (p=0 or p=1) and symmetry.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
Clarify the assumptions (e.g., constant per-game win probability, independence) and then outline a simulation that repeatedly plays best-of-5 series, recording the number of games. Finally, estimate the probability as the proportion of series that last exactly 5 games, and discuss how to choose the number of simulations for a desired precision.
Pro tip: Mention that you can compute the exact probability analytically (e.g., 2 * p^2 * (1-p)^2 * (something) or via negative binomial) and use that to validate your simulation—this shows you understand the problem deeply and can sanity-check results.
State that each game is independent and has a constant probability p of the stronger team winning (or assume p=0.5 if not specified). Define what 'lasts exactly 5 games' means: the series is tied 2-2 after 4 games, and the 5th game decides the winner.
For each simulated series, simulate games until one team reaches 3 wins. Count the number of games played. Repeat this for N independent series.
Calculate the proportion of series that lasted exactly 5 games. Also compute a confidence interval (e.g., using the normal approximation) to quantify the Monte Carlo error.
Compare the simulation result to the analytical probability (if p is known) to ensure correctness. Discuss how the probability depends on p and the implications for the business context.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
Start by clarifying that the matches are independent Bernoulli trials with a constant probability of success, then present the standard Wald interval using the sample proportion and its standard error. Also discuss the limitations of the Wald interval for small M or extreme probabilities, and mention more robust alternatives like the Wilson score interval or Clopper-Pearson exact interval.
Pro tip: In practice, especially for A/B testing at scale, the Wilson score interval is preferred because it maintains good coverage even for small sample sizes or probabilities near 0 or 1, and it's easy to compute.
Confirm that the M matches are independent and identically distributed Bernoulli trials with a constant probability of success p. State that the estimate is the sample proportion p_hat = X/M, where X is the number of successes.
Select an appropriate method based on M and p_hat. For large M and p_hat not near 0 or 1, the Wald interval is simple. For small M or extreme p_hat, use Wilson score or Clopper-Pearson exact interval.
For the Wald interval, calculate p_hat ± z_{α/2} * sqrt(p_hat*(1-p_hat)/M). For Wilson, use the formula that adjusts for continuity and skewness. For Clopper-Pearson, use the beta distribution quantiles.
Explain that the interval provides a range of plausible values for the true probability, with the chosen confidence level. Discuss any caveats about coverage probability and the impact of M.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.