2025 ACJC P2 Q9

A fruit seller sources his oranges from a supplier with a large plantation where \(1\)% of the oranges are rotten. The supplier picks the oranges regardless of whether the oranges are rotten or not, and packs \(20\) oranges in each box at random.

1. The random variable \(X\) denotes the number of rotten oranges in a randomly chosen box. State in context two assumptions for \(X\) to follow a binomial distribution.[2]

Assume that \(X\) follows a binomial distribution for the rest of this question. The fruit seller considers a box containing between \(2\) and \(5\) rotten oranges inclusive as substandard.

2. Find the probability that a randomly chosen box is substandard.[1]
3. Find the expected number of rotten oranges in a box.[1]

A truckload of \(10\) randomly chosen boxes of oranges is then sent to the fruit seller.

4. Find the probability that not more than \(2\) boxes from the entire truckload are substandard.[2]

Upon receiving the oranges, the fruit seller inspects the boxes with these conditions:

1. Accept the entire truckload if the first box opened has no rotten oranges.
2. Reject the entire truckload if the first box opened has more than \(1\) rotten orange.
3. If the first box has exactly \(1\) rotten orange, the fruit seller opens a second box.
4. If the second box has at most \(1\) rotten orange, he will accept the entire truckload. Otherwise, he will reject the entire truckload.

5. Find the probability that he will reject the entire truckload.[2]