Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 36 / 1 / c / Solution

Conditional on the success probability, the count has binomial distribution. Integrating its likelihood under the uniform prior distribution, or equivalently a beta distribution with parameters $(1,1)$, givesThe prior predictive distribution is therefore uniform over the possible counts. Excluding the single point $\theta =1/2$ does not alter this integral because a continuous prior distribution gives that point zero probability. This calculation is the marginalization underlying Beta-binomial conjugacy.