 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2008/iii/paper-47/4/b/solution

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

Since the original observations are distinct, a bootstrap sample has no repeated value precisely when it contains all

n

original values once. Of the

n^{n}

equally likely ordered resamples,

n!

are permutations without repetition. Therefore

P (at least one repeated value) = 1 - \frac{n !}{n ^{n}} .

The probability is zero for

n = 1

and rapidly approaches one as

n

grows.

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