2026 HCI P2 Q9

A football academy uses a penalty-taking test to scout new male players. For a typical candidate, the probability of scoring a goal in a single penalty is \(p\). The number of goals scored in a set of \(n\) penalties is denoted by \(X\).

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

Assume now that \(X\) follows a binomial distribution.

2. The probability of a candidate, Albert, scoring a goal in a single penalty is 0.65. Albert takes a set of 15 penalties. Find the most likely number of goals scored by Albert.[2]
3. The probability of another candidate, Bob, scoring a goal in a single penalty is \(p\). It is known that in a set of 10 penalties, the most likely number of goals scored by Bob is 3. Find the range of values of \(p\).[3]

To be recruited, a candidate must pass a Two-Stage Trial:

• Stage 1: A candidate takes a set of 10 penalties. If he scores more than 7 goals, he is recruited immediately.
• Stage 2: If a candidate scores 5 to 7 goals in a set of 10 penalties in Stage 1, he takes another set of 10 penalties. If he scores more than 5 goals in the second set of penalties, he is recruited.
• Otherwise, he is not recruited.

4. A candidate is known to have been recruited. If \(p=0.6\), calculate the probability that he was recruited only after Stage 2.[4]
5. After 100 candidates completed Stage 1 of their trial, the academy analysed the distribution of the number of goals scored out of a set of 10 penalties. It was found that this distribution has two modes — one at 3 goals and one at 8 goals. Discuss whether this observation is consistent with the population following a single binomial distribution.[1]