2023 ACJC P2 Q6

A factory manufactures a large number of chips in a variety of colours and packs them into boxes. Each box contains \(36\) randomly chosen chips. On average, \(30\%\) of the chips are yellow. You may assume that the number of yellow chips in a box follows a binomial distribution.

1. A randomly chosen box is found to have at most \(9\) yellow chips. Find the probability that it contains more than \(4\) yellow chips.[2]

\(75\) randomly chosen boxes are packed into a carton for shipment.

2. Find the probability that the \({{60}^{\mathrm{th}}}\) box chosen is the \({{14}^{\mathrm{th}}}\) box with at most \(9\) yellow chips.[2]
3. \(30\) cartons are sold to an arcade. Using a suitable approximation, find the probability that the mean number of boxes containing at most \(9\) yellow chips in each carton is more than \(25\).[3]