2023 NYJC P2 Q8

On average, the probability that a student from a randomly chosen class answers the first question of a multiple-choice examination correctly is \(p\). The number of students in a class of \(30\) who gets the first question correct is denoted by the random variable \(X\).

It is given that \(X\) can be modelled by a binomial distribution.

1. It is given that the most probable number of students in the class who will get the first question correct is \(6\). By considering expressions in terms of \(p\), find exactly the possible range of values of \(p\).[4]

For the rest of the question, let \(p=0.2\).

2. Find the probability that more than \(5\) but at most \(10\) students answered the first question correctly.[2]
3. Forty classes, each with \(30\) students, are randomly selected. By using a suitable approximation, find the probability that the average number of students who answer the first question correctly per class is at least \(5\).[3]