Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 40 / 3 / i / Solution

Treat the supplied uniform variates as independent $U(0,1)$ draws, as required for Monte Carlo simulation. Use $n$ of them and returnEach indicator has a Bernoulli distribution of parameter $p$, and the indicators are independent. For any subset of $r$ successes the probability is $p^r(1-p{)}^{n-r}$; there are $\left(\genfrac{}{}{0ex}{}{n}{r}\right)$ such subsets. Thus $\mathbb{P}(X=r)=\left(\genfrac{}{}{0ex}{}{n}{r}\right)p^r(1-p{)}^{n-r}$, the binomial distribution. The construction includes the cases $p=0,1$ and $n=0$.