2021 EJC P2 Q10

In this question, you should state the parameters of any distributions that you use.

A stationery factory manufactures pens for sale. The diameters (in mm) of the pens have distribution \(\mathrm{N}\left( \mathrm{10,0}\mathrm{.003} \right)\). The pens are packed in sets of \(24\) into a box. Within the box, the pens are laid out side by side. The widths of the boxes (in mm) are normally distributed with mean \(\mathrm{240}\mathrm{.5}\)mm and variance \(\mathrm{0}\mathrm{.02}\)mm\(^{2}\).

1. Find the probability that a random sample of \(24\) pens would fit into a randomly selected box.[3]

A pen is considered “defective” if its diameter is not within \(0.1\) mm of the population mean.

2. Find the probability that a randomly selected pen is defective.[2]

Pens are produced and inspected in batches. For each batch, a sample of \(12\) pens is randomly selected and checked.

• If there are fewer than \(2\) defective pens in this sample of \(12\), the batch passes the inspection.

• If there are exactly \(2\) defective pens in this sample, a second sample of \(12\) pens is randomly selected and checked. If there are no defective pens in the second sample, the batch passes the inspection.

• Otherwise, the batch does not pass the inspection.

3. Show that the probability of a batch of pens passing the inspection is \(\mathrm{0}\mathrm{.871}\), correct to \(3\) significant figures.[3]
4. Find the probability that not more than \(3\) defective pens were found in a batch of pens during the inspection process, given that the batch of pens did not pass the inspection.[3]

The factory also manufactures and packs pencils and crayons in boxes of \(12\) each. During a promotion period, special pencils and special crayons are produced and randomly included in some boxes. The number of special pencils in a box of pencils has distribution \(\mathrm{B}\left( 12,0.07 \right)\), while the number of special crayons in a box of crayons has distribution \(\mathrm{B}\left( \mathrm{12, 0}\mathrm{.06} \right)\).

5. \(40\) boxes of pencils and \(40\) boxes of crayons are randomly selected. Find the probability that the mean number of special pencils per box of pencils is more than the mean number of special crayons per box of crayons.[4]