N2017 P2 Q9

On average \(8\%\) of a certain brand of kitchen lights are faulty. The lights are sold in boxes of \(12\).

1. State, in context, two assumptions needed for the number of faulty lights in a box to be well modelled by a binomial distribution.[2]

Assume now that the number of faulty lights in a box has a binomial distribution.

2. Find the probability that a box of \(12\) of these kitchen lights contains at least \(1\) faulty lights.[1]

The boxes are packed into cartons. Each carton contains \(20\) boxes.

3. Find the probability that each box in one randomly selected carton contains at least one faulty light.[1]
4. Find the probability that there are at least \(20\) faulty lights in a randomly selected carton.[2]
5. Explain why the answer to part (iv) is greater than the answer to part (iii).[1]

The manufacturer introduces a quick test to check if lights are faulty. Lights identified as faulty are discarded. If a light is faulty there is a \(95\%\) chance that the quick test will correctly identify the light as faulty. If the light is not faulty, there is a \(6\%\) chance that the quick test will incorrectly identify the light as faulty.

6. Find the probability that a light identified as faulty by the quick test is not faulty.[3]
7. Find the probability that the quick test correctly identifies lights as faulty or not faulty.[1]
8. Discuss briefly whether the quick test is worthwhile.[1]