A coin has an unknown bias p, modelled as uniform on [0,1]. You flip it 10 times and see 10 heads. What is the probability the next flip is a head?

A coin has an unknown bias p, modelled as uniform on [0,1]. You flip it 10 times and see 10 heads. What is the probability the next flip is a head?

Approach: Update the uniform prior with the binomial likelihood to get a Beta posterior, then take its mean, which is the predictive probability for one further flip.

11/12. With a uniform prior the beta posterior after 10 heads and 0 tails is Beta(11, 1), with density 11 p^10 on [0,1]. The predictive probability of another head is the posterior mean E[p] = 11/12 = 0.9167. This is the rule of succession, (s+1)/(n+2) with s = n = 10. A maximum likelihood estimate would report 1, which asserts that a tail is now impossible, and the Beta posterior avoids that by carrying the prior's one pseudo-head and one pseudo-tail. Ten heads is strong evidence yet still leaves an 8.3% chance of a tail next.

Follow-up: How many further heads are needed before the predictive probability exceeds 0.99?

Key concepts: beta posterior, uniform prior, rule of succession, posterior mean.