2021 NJC P2 Q8

National Fruit Company owns a large tomato farm. The tomatoes produced are harvested and sold in boxes of \(25\). It is known that \(100p\%\) of the tomatoes are rotten. For these boxes, the mean number of rotten tomatoes in a box is \(1\).

1. Explain why the context above may not be well-modelled by a binomial distribution.[1]

Assume now that the context above is well-modelled by a binomial distribution.

2. State the value of \(p\).[1]
3. Find the probability that a box chosen at random has less than \(2\) rotten tomato.[2]
4. A customer chose a box and inspected the contents individually. Find the probability that the twenty-first tomato is the fourth rotten tomato and no rotten tomatoes are found subsequently.[3]

Boxes that contains at least \(24\) tomatoes that are not rotten are deemed satisfactory.

5. A customer first picks \(3\) boxes of tomatoes, of which at least \(2\) boxes are satisfactory. The customer then decides to buy another \(5\) boxes. Find the probability that exactly \(6\) of the \(8\) boxes are satisfactory.[4]