2025 RVHS P2 Q9

A machine produces a large number of camera lenses daily. During the process, lenses could get scratched and become defective.

1. State, in the context of the question, one assumption needed for the number of defective lenses made to be modelled by a binomial distribution and explain why the assumption may not hold in this context.[2]

It is assumed that the number of defective lenses produced daily has a binomial distribution and that on average, \(1\) in \(20\) lenses produced are defective.

2. \(30\) independent samples of \(100\) lenses each are taken. Find the probability that there are more than \(10\) samples in which there are exactly \(3\) defective lenses in each sample.[2]
3. \(2\) independent samples of \(100\) lenses are taken. It is given that exactly one of the samples was found to have exactly \(3\) defective lenses. Find the probability that there are a total of \(10\) defective lenses in the \(2\) samples.[2]
4. Every day, lenses are randomly selected for quality checks on the production. An initial sample of \(100\) lenses is obtained.
   • The entire production of lenses for that day would be deemed to have passed the quality check when there are at most \(2\) defective lenses in the initial sample.
   • If there are more than \(3\) defective lenses found in the initial sample, then the entire production of lenses for that day will be rejected.
   • If the initial sample contains exactly \(3\) defective lenses, another sample of \(100\) lenses will be taken. If the \(2\) samples contain an average of at most \(2\) defective lenses, then the entire production of lenses for that day would be deemed to have passed the quality check.
   Find the probability that the production of lenses of any particular day would have passed the quality check.[3]