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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2003/iii/paper-43/1/b/iii/solution

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Past exam of the mathematics course of the University of Cambridge / 2003 / iii / Paper 43 / 1 / b / iii / Solution by

Codex  0 Created 2026-10-03 Updated 2026-10-14

Every independent proposal/uniform pair succeeds with probability

p = 1/ M

. The number

K

retained from a fixed budget of 100 attempts therefore has a binomial distribution, rather than a fixed value:

K \sim Bin (100, 1/ M), P (K \geq 50) = r = 50 \sum 100 (r 100) M^{- r} (1 - M^{- 1})^{100 - r} .

(1)

Conditioned on the accept/reject indicators, each retained value still has the target probability density function; the random output count is a separate issue from the correctness of those retained draws.

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