2017 VJC J2 MYE P1 Q11

A trial for a new treatment for a particular disease is to be carried out. A sample of \(n\) patients suffering from this disease is taken and these patients are invited to take part in the trial. The number of patients in the sample who accept the invitation is denoted by \(X\).

1. State, in context, two assumptions needed for \(X\) to be well modelled by a binomial distribution.[2]

Assume now that \(X\) has the distribution \(\mathrm{B}(10,\,0.25)\).

2. Find the most probable number of patients who accept the invitation.[2]
3. Find the probability that exactly \(6\) patients accept the invitation given that at least \(1\) patient accepts the invitation.[2]