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

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

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

For the evolved-companion fraction, a red giant has another evolved component precisely for unordered

GG

G W

pairs. By conditional independence at fixed age, their combined slice probability is

g^{2} + 2 g w = 9 g^{2}

. Therefore

Pr (GG or G W) = \int_{0}^{1} \frac{9 y ^{2}}{10000} d y = \frac{9}{30000} .

(1)

Divide by the system fraction containing at least one red giant to obtain the requested conditional fraction:

Pr (another evolved component ∣ at least one G) = \frac{9}{299} ≃ 3.01003%.

(2)

A double-giant system is counted once in this denominator, and the factor two for a giant–white dwarf pair represents its two possible component orderings.

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