2024 RI H1 Q8

A manufacturer produces light bulbs in batches. Historically, \(1.6\) % of the bulbs produced are defective.

A quality control inspector randomly selects a sample of \(20\) bulbs from a batch for inspection.

1. Find the probability that at most one bulb in the sample is defective.[2]

The inspector marks a batch as “passed” if at most one bulb in the sample of \(20\) is defective. Otherwise, he randomly selects a new sample of \(10\) bulbs from the same batch. If there is no defective bulb in this new sample of \(10\) bulbs, the batch is marked as “passed”. Otherwise, the batch is marked as “failed”.

2. Show that there is less than \(0.6\) % chance that a batch is marked as “failed”.[4]

The inspector inspects \(800\) batches of bulbs in a month.

3. Find the expected number of batches of bulbs that will be marked as “passed”.[2]