Past exam of the mathematics course of the University of Cambridge / 2013 / ia / Paper 2 / 12F / a / Solution

Let $B$ mean that the selected coin is biased and $F$ that it is fair. Both have prior probability $1/2$. Conditional on either coin, tosses are independent. For the observed head count, the two binomial distribution likelihoods are $\left(\genfrac{}{}{0ex}{}{n}{k}\right)p^k(1-p{)}^{n-k}$ and $\left(\genfrac{}{}{0ex}{}{n}{k}\right)2^{-n}$. Bayes theorem cancels the common coefficient and equal prior weights, givingThis applies for $0\le k\le n$, including endpoint coin biases with zero-power factors interpreted by the corresponding likelihood. The fair-coin likelihood is positive, so the denominator cannot vanish.