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ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2022/discrete-mathematics-continuous-mathematics-and-probability/probability/7/b/iii/solution

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Past exam of the year 1 of the computer science course of the University of Oxford / 2022 / Discrete mathematics, continuous mathematics, and probability / Probability / 7 / b / iii / Solution by

Codex  0 2026-10-25

Write

p_{n} = (1 - 1/ n)^{n}

, the probability that all balls avoid a specified box. Each of the empty-box indicators in uniform allocation is an indicator random variable with success probability

p_{n}

. Since

I_{i}^{2} = I_{i}

, its expectation and variance are

E I_{i} = p_{n}, var (I_{i}) = p_{n} (1 - p_{n}) .

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