2021 HCI P2 Q7

A toy factory manufactures gel beads which are polymer beads that increase in size when soaked in water. On average, \(8\%\) of the gel beads are defective. The gel beads are packed in bags of \(500\). A significant number of customers recently gave feedback that many of the gel beads they bought could not expand in water or cracked while expanding. The quality control department decides to take a random sample of \(20\) gel beads from each bag to test. If more than \(4\) gel beads are found to be defective in the sample of \(20\), the bag is rejected. Otherwise the bag is accepted.

1. State, in context, two assumptions needed for the number of defective gel beads in the sample to be well modelled by a binomial distribution.[2]

Assume now that the number of defective gel beads in a sample of \(20\) is modelled by a binomial distribution.

2. Find the probability that a randomly chosen bag of gel beads is rejected.[1]
3. An officer from the quality control department is in charge of inspecting \(10\) randomly chosen bags of gel beads. Find the probability that the last bag inspected is the second bag that is being rejected.[2]
4. A random sample of \(20\) gel beads is taken from a particular bag. Given that the bag is rejected, find the probability that there are more than \(13\) gel beads with no defects in the random sample of \(20\) gel beads.[3]
5. The quality control department now decides to test \(50\) randomly chosen bags of gel beads. Find the probability that the mean number of defective gel beads found in the sample of each bag will not exceed \(1.5\).[2]