2019 SAJC P2 Q10

A factory manufactures a large number of erasers in a variety of colours. Each box of erasers contains \(36\) randomly chosen erasers. On average, \(20\)% of erasers in the box are blue.

1. State, in context, two assumptions needed for the number of blue erasers in a box to be well modelled by a binomial distribution.[2]
2. Find the probability that a randomly chosen box of erasers contain at most six blue erasers.[1]

\(200\) randomly chosen boxes are packed into a carton. A carton is considered acceptable if at least \(40\)% of the boxes contain at most six blue erasers each.

3. Find the probability that a randomly chosen carton is acceptable.[3]

The cartons are exported by sea. Over a one-year period, there are \(30\) shipments of \(150\) cartons each.

4. Using a suitable approximation, find the probability that the mean number of acceptable cartons per shipment for the year is less than \(80\).

The owner decided to change the proportion of blue erasers to \(p\). A box of erasers is chosen.

5. Write down in terms of \(p\), the probability that the box contains exactly one blue eraser.[1]
6. The probability that a box contains exactly one blue eraser is twice the probability that the box contains exactly two blue erasers. Write an equation in terms of \(p\), and hence find the value of \(p\).[2]