2025 EJC H1 Q9

At the fruits section of the office canteen at Aonyx corporation, employees select a fruit as part of their lunch. On average, \(25\)% of the employees will select an apple. A random sample of \(n\) employees are interviewed on their preferences and the number of employees who will select an apple is denoted by \(X\).

1. State in context of the question, an assumption needed to model \(X\) by a binomial distribution.[1]

Assume now that \(X\) has the distribution \(\mathrm{B}(n, 0.25)\).

2. Find the least value of \(n\) such that the probability of at least one employee selecting an apple is at least \(0.98\).[2]

For the remaining parts of the question, use \(n = 40\).

3. Find the probability that no more than \(24\) of the employees will select other fruits.[2]
4. Find the probability that at most \(20\) of the employees will select an apple and the last employee interviewed will select an apple, giving your answer correct to \(5\) decimal places.[3]