February/March 2026 Paper 62

A shop sells two types of cell phone. Sales of the two types occur at constant average rates of 3.2 per week for one type and 2.9 per week for the other type. You may assume that sales occur at random and independently of one another. A week is chosen at random. Find the probability that the shop sells a total of 5 cell phones.

Find the probability that the shop sells exactly 5 phones of one type and none of the other type.

Use a suitable approximating distribution to find the probability that the shop sells a total of more than 30 cell phones in 6 randomly chosen weeks.

Justify your approximating distribution.

The weekly sales, $X$ and $Y$, of two types of computer, have the distributions $X\sim \mathrm{Po}(\lambda )$ and $Y\sim \mathrm{Po}(\mu )$. Given that $\mathrm{P}(X=0)=2\mathrm{P}(Y=1)$, find an expression for $\lambda$ in terms of $\mu$.