N2025 P2 Q9

A company produces a variety of domestic appliances. One of Alan's jobs is to test a fixed number of randomly chosen refrigerators each day to see if any are faulty. The number found to be faulty each day 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 \(\mathrm{B}(90, 0.02)\).

2. Find the probability that, on a randomly chosen day, Alan finds more than one faulty refrigerator.[2]
3. Find the probability that, in a randomly chosen \(5\)-day working week, Alan finds more than one faulty refrigerator on fewer than two days.[2]
4. Find the probability that, in a randomly chosen \(5\)-day working week, Alan finds fewer than \(10\) faulty refrigerators in total.[2]

The company also makes washing machines. Alan also tests a fixed number of randomly chosen washing machines each day. The number of washing machines found to be faulty each day is denoted by \(Y\). The distribution of \(Y\) is \(\mathrm{B}(60, 0.03)\). The random variables \(X\) and \(Y\) are independent.

5. Find the probability that, on a randomly chosen day, the total number of refrigerators and washing machines Alan finds to be faulty is exactly two.[3]