Past exam of the mathematics course of the University of Oxford / 2023 / Prelims / Mathematics III / 4 / b / ii

Table of contents
- Solution 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

or

r = n

already have zero probability. These are the transition probabilities of the Bernoulli-Laplace diffusion model.

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