2013 TJC P2 Q12

A factory produces a particular type of electronic component. The probability of a component being acceptable is \(0.96\). The components are packed in boxes of \(24\). A box is rejected if it contains fewer than \(22\) acceptable components.

1. Show that the probability that a randomly chosen box is rejected is \(0.0693\).[1]
2. \(5\) boxes are chosen at random. Find the probability that two of the boxes each contain exactly \(22\) acceptable components and one box is rejected.[3]

The factory produces \(70\) boxes each day. The production is being monitored for \(n\) days, where \(n\) is large.

3. Find the expected number of rejected boxes in a day.[2]