Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2020 / Paper 1 / 16 / b / ii / Solution

Write $N={\sum }_{j=1}^{10}{X}_j$, where the independent Bernoulli random variables indicate rottenness. Since $X_j^2=X_j$, their variances are $E[X_j^2]-E[X_j{]}^2=p-p^2$. The independence makes all cross covariances zero, soThis standard deviation vanishes at $p=0$ and $p=1$, when the count is certain.