This is a Bayes' theorem question and I almost answered it like a frequentist, which would've been completely wrong.
Clarify that the question is about inferring the probability the coin is fair given the observed data, not the probability of the data itself. Use Bayesian reasoning: start with a prior belief about fairness, compute the likelihood of 10 heads under a fair coin, and update to a posterior. Alternatively, discuss hypothesis testing and p-values, but emphasize that a p-value does not directly give the probability the coin is fair.
Pro tip: Acknowledge that the answer depends on your prior and that in a real-world scenario, you'd also consider alternative hypotheses (e.g., biased coin, two-headed coin). This shows you understand the nuances of statistical inference and avoid giving a naive frequentist answer.
Restate the problem: Given 10 heads in a row, what is the probability the coin is fair? Distinguish between the probability of observing the data under a fair coin and the probability the coin is fair given the data.
Decide whether to use Bayesian inference (natural for 'probability the coin is fair') or frequentist hypothesis testing. Explain the choice and its implications.
Define a prior probability that the coin is fair (e.g., 0.5 or based on domain knowledge). Compute the likelihood of 10 heads under a fair coin (0.5^10 = 1/1024) and under an alternative (e.g., biased coin with probability p of heads). Use Bayes' theorem to compute the posterior probability that the coin is fair.
If using hypothesis testing, set null hypothesis H0: coin is fair (p=0.5). Compute the p-value: probability of observing 10 heads or more extreme under H0, which is 2 * 0.5^10 = 1/512 ≈ 0.00195. Conclude that if p-value < alpha, reject H0, but note this does not give the probability the coin is fair.
Highlight that the Bayesian result depends heavily on the prior and the specification of alternative hypotheses. Mention that with only 10 flips, the evidence is not extremely strong, and in practice, you'd gather more data or consider other factors (e.g., physical inspection of the coin).
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.