Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2005 / Paper 1 / 3

Let $G$ mean that the tested person is guilty and $M$ mean a positive match. Under the specified model, $P(G)=1/N$, $P(M\mid G)=1$ and $P(M\mid G^c)=p$. Bayes' theorem givesThe posterior probability for a unique source after a positive test depends on the number of alternative possible sources as well as the false-positive probability. In particular, the false-positive probability $p$ is not the probability of innocence conditional on a positive match.

Under the independent uniform-birthday model, condition on the first person's birthday. Exactly one of the second person's $365$ equally likely dates matches, soThis concerns one selected pair; the birthday problem allowing any pair among all guests is a different event.

For matching one specified birthday, each of the $N-1$ other guests avoids the host's date with probability $364/365$. Independence givesFor this to exceed $1/2$, the negative logarithm of $364/365$ reverses the inequality on division, giving $N-1>\ln (1/2)/\ln (364/365)\simeq 252.652$. Thus the smallest total attendance is $N=254$, including the host.

The quoted pass probabilities on later attempts are conditional on reaching those attempts. If $K$ is the first successful attempt, thenThey sum to $99/100$, so conditioning on success by the third attempt is essential. The conditional mean of a bounded success time isThe remaining probability $1/100$ represents failing all three attempts and is excluded by this conditioning.