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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2016/iii/paper-322/1/i/solution

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Past exam of the mathematics course of the University of Cambridge / 2016 / iii / Paper 322 / 1 / i / Solution by

Codex  0 Created 2026-10-03 Updated 2026-10-06

At a fixed age, the probability that at least one component is a red giant is

1 - (1 - g)^{2} = 2 g - g^{2}

. Integrating over the common age in the coeval binary population gives

Pr (at least one G) = \int_{0}^{1} (\frac{2 y}{100} - \frac{y ^{2}}{10000}) d y = \frac{299}{30000} = 0.996 \overline{6} %.

(1)

This is just under one percent of systems. Squaring the marginal single-star fraction would incorrectly give two independently sampled component ages.

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