Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2009 / Paper 1 / 12 / a

Neither biscuit type can run out in this case. Treat each freely made choice as an independent Bernoulli trial, with success meaning chocolate. The number taken after $n$ choices has a binomial distribution, soThe binomial coefficient counts the possible success positions, and every such arrangement has the same probability. Independence of successive choices is the probabilistic model implicit in the question; identical marginal preferences alone would not imply it.

Write the chocolate count as $K_n={\sum }_{j=1}^n{I}_j$ with independent Bernoulli random variables. Each has expected value $p$ and variance $p(1-p)$. By linearity of expected value and additivity of variance for independent variables,