 Solution

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

R_{k}

. Since the conditional mean is affine,

r_{k + 1} = E [E [R_{k + 1} ∣ R_{k}]] = 1 + (1 - \frac{2}{n}) r_{k} .

Here

r_{0} = n

. This uses linearity of expectation and does not assert independence of successive urn counts.

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