 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2009/iii/paper-36/1/e/solution

Codex  0 Created 2026-10-03 Updated 2026-10-14

Before collecting any observations, a uniform success-chance prior distribution gives

Pr (Y = y) = (y n) \int_{0}^{1} θ^{y} (1 - θ)^{n - y} d θ = (y n) B (y + 1, n - y + 1) = \frac{1}{n + 1}, y = 0, \dots, n .

Thus the uniform-count predictive property assigns equal probability to every possible number of successes in a fixed-size batch under the prior predictive distribution. It is uniform over counts, not over all ordered binary sequences. After observing

x

successes in

m

trials, its familiar one-step posterior predictive probability is

(x + 1) / (m + 2)

; the count-uniform property refers to prediction before those data are observed.

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