2024 JPJC P2 Q8

A store owner receives a shipment of stationery items, including notebooks, pens, and correction tapes. Historical data indicates that \(1\%\) of the notebooks, \(2\%\) of the pens, and \(4\%\) of the correction tapes are defective. The quality of notebooks, pens and correction tapes is independent of one another. The store owner decides to sell the stationery in \(180\) packets, each containing one notebook, two pens and one correction tape. A packet is deemed unsatisfactory if any of the four items is defective.

1. Show that the probability that a randomly selected packet is unsatisfactory is \(0.0872\), correct to \(3\) significant figures.[2]

The number of packets that are unsatisfactory is denoted by \(X\). You may assume that \(X\) can be modelled by a binomial distribution

2. Find the probability that there are at least \(5\) but less than \(10\) packets that are unsatisfactory.[2]
3. Find the least value of \(r\) such that the probability that there are more than \(r\) packets that are unsatisfactory is at most \(0.12\).[2]

Before selling the packets of stationery, he decided to select a sample of \(9\) packets to check for unsatisfactory packets.

4. How should the packets be selected? Give a reason for this method of selection.[2]
5. Find the probability that the ninth packet is the third unsatisfactory packet selected.[2]