2024 CJC P2 Q7

Aran is a telemarketer working in a large city. Each day, he speaks to \(50\) potential customers and on average, he successfully makes a sale \(6\%\) of the time. The number of successful sales each day is denoted by the random variable \(T\).

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

Assume now that \(T\) has a binomial distribution.

2. Find the probability that Aran makes at least \(5\) successful sales on a randomly chosen day.[2]
3. Aran receives a bonus if he makes at least \(5\) successful sales a day. Find the probability that he receives a bonus on \(2\) days out of a randomly chosen \(5\)-day work week.[2]

Aran is part of a team of \(100\) similar telemarketers.

4. Estimate the probability that on a randomly chosen day, the team makes more than \(3.5\) successful sales on average.[3]