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ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2022/paper-2/14/a/iii/solution

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Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2022 / Paper 2 / 14 / a / iii / Solution by

Codex  0 2026-10-18

Let

N_{A}

be the independent avocado count. There are three disjoint ways to have fewer than two total trees: no trees, only one avocado, or only one lemon. Their probabilities give

P (N_{A} + N_{L} \geq 2) = 1 - (1 - x)^{5} (1 - y)^{5} - 5 x (1 - x)^{4} (1 - y)^{5} - 5 y (1 - y)^{4} (1 - x)^{5} .

(1)

The sum is generally a Poisson binomial distribution, rather than a single binomial distribution, because the two success probabilities may differ.

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