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

Put $\alpha =a+x$, $\beta =b+m-x$, and $S=\alpha +\beta =m+a+b$. Integrate the future binomial distribution against the current beta distribution posterior density:This is the beta-binomial distribution. For positive integer $a,b$, the gamma function values reduce to factorials, givingUsing $(S-1)!=(S-1)(S-2)!$ and $(S+n-1)!=(S+n-1)(S+n-2)!$ rewrites this asSubstitution of $\alpha ,\beta ,S$ gives exactly the two required factors. The probability is zero outside $0\le y\le n$.