Solution
The Two Coins
Show the problem again
A bag contains two coins: one fair, one with heads on both sides. You pick one uniformly at random and flip it three times, getting heads all three times. What is the probability you picked the two-headed coin? Express as a fraction.
Worked solution
The answer is 8/9. By Bayes' theorem: P(HHH | two-headed) = 1 and P(HHH | fair) = 1/8, with equal priors. So P(two-headed | HHH) = 1/(1 + 1/8) = 8/9. Each additional head multiplies the evidence in favor of the biased coin by 2: a clean illustration of likelihood-ratio updating.
Source: Standard Bayesian coin-identification exercise from introductory probability texts. Statement written for AxiomIQ.