Seemed straightforward at first and I almost rushed through it.
Start by clarifying the number of flips and the specific outcome of interest, then model the process as a binomial distribution with p=0.5. Explain the assumptions (independence, fair coin) and derive the probability mass function, mean, and variance, and discuss how the distribution changes as the number of flips increases.
Pro tip: Connect the binomial distribution to the normal approximation for large n, and mention the law of large numbers to show how the proportion of heads converges to 0.5. This demonstrates both statistical maturity and practical insight for product analytics.
Ask the interviewer to specify the number of flips (n) and whether they want the distribution of the number of heads, the proportion of heads, or the sequence of outcomes. This ensures you address the exact ask.
Let X be the number of heads in n flips. State that each flip is independent and has probability p=0.5 of heads, so X follows a binomial distribution: X ~ Binomial(n, 0.5).
Write the PMF: P(X = k) = C(n, k) * (0.5)^k * (0.5)^(n-k) = C(n, k) / 2^n. Explain that this gives the probability of exactly k heads in n flips.
Calculate the mean (np = n/2) and variance (np(1-p) = n/4). Note that the distribution is symmetric around n/2 and becomes more bell-shaped as n increases.
For large n, apply the normal approximation (or Central Limit Theorem) to approximate probabilities. Mention the law of large numbers: the proportion of heads converges to 0.5, and the standard deviation of the proportion shrinks as 1/sqrt(n).
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.