 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-oxford/2023/prelims/mathematics-iii/4/b/ii/solution

Conditional on

R_{k} = r

, urn 1 has

r

red and

n - r

blue balls, while urn 2 has

n - r

red and

r

blue balls. The independent selections therefore give

P (R_{k + 1} = r + 1 ∣ R_{k} = r) P (R_{k + 1} = r - 1 ∣ R_{k} = r) P (R_{k + 1} = r ∣ R_{k} = r) = (n - r)^{2} / n^{2}, = r^{2} / n^{2}, = 2 r (n - r) / n^{2} .

Every other conditional probability is zero. The out-of-range terms at

r = 0

r = n

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