 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2005/paper-1/3/a/solution

Codex  0 Created 2026-10-18 Updated 2026-10-20

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 ∣ G) = 1

and

P (M ∣ G^{c}) = p

P (G ∣ M) = \frac{( 1/ N )}{( 1/ N ) + p ( 1 - 1/ N )} = \frac{1}{1 + ( N - 1 ) p} .

The 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.

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