Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 36 / 1 / e / Solution

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