2025 YIJC P2 Q11

In a large shipment of second-hand T-shirts, \(4\)% of the T-shirts are torn. The T-shirts are sold in boxes of \(25\) pieces each. Let \(X\) denote the number of torn T-shirts in a box.

1. State, in the context of the question, two assumptions needed to model \(X\) by a binomial distribution.[2]

You are now given that \(X\) can be modelled by a binomial distribution.

2. A box is randomly chosen. Find the probability that a box contains at least \(2\) torn T-shirts.[2]
3. A box is deemed to be of inferior quality if it contains at least \(2\) torn T-shirts. Find the probability that, in a random sample of \(10\) boxes of T-shirts, more than \(3\) boxes are of inferior quality.[2]

A distributor purchases a batch of \(60\) boxes of T-shirts.

4. Find the probability that the average number of torn T-shirts per box is more than \(1.1\).[3]
5. Find the most probable number of boxes of inferior quality in the batch.[2]

In a large shipment of second-hand blouses, a proportion \(p\) of the blouses is torn. The blouses are also sold in boxes of \(25\) pieces each. The number of torn blouses in a box follows a binomial distribution. It is known that the probability of a box containing exactly \(2\) torn blouses is \(0.206\).

6. Write down an equation satisfied by \(p\). Hence, find the value of \(p\), given that \(p > 0.1\).[2]