The next flip after a hundred straight heads

Question

A coin has just landed heads on all 100 of its tosses. What is your probability that the 101st toss is also heads?

Short answer

Essentially 1, unless you are told the coin is certainly fair. If fairness is guaranteed, tosses are independent and the answer is $\tfrac12$ . But a fair coin produces 100 heads with probability $2^{-100}\approx 8\times10^{-31}$ , so under any sensible prior that allows even a tiny chance of a biased or two-headed coin, Bayes' rule puts almost all posterior weight on a heads-only coin. Interviewers want you to state both views and argue for the Bayesian one.

Adapted from Crack, Heard on the Street, Q5.5.10, p.55; question and answer rewritten by SelfMadeQuant.com.

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