2025 NJC P2 Q9

In a sales campaign, a supermarket gives each shopper a receipt printed with a cartoon character when a shopper pays for his or her purchase. There are \(10\) different cartoon characters available, character \(A, B, C, \dots, J\). The printed cartoon character is equally likely to be any one of the \(10\) different cartoon characters.

A shopper received three receipts after having made three purchases.

1. Find the probability that only one receipt has the character \(A\) printed on it.[2]
2. Given that exactly one of the cartoon characters printed is \(A\), find the probability that the other two characters printed are \(B\) and \(C\).[3]

To redeem a gift, exactly \(8\) distinct cartoon characters need to be collected. After some time, Tom collected \(7\) distinct characters. Let \(p_n\) be the probability that Tom can redeem the gift on exactly the \(n^{\mathrm{th}}\) additional purchase.

3. Find \(p_1\) and the recurrence relation in the form \(p_n = \mathrm{f}(p_{n-1})\), for \(n \ge 2\).[2]
4. Find the probability, in terms of \(n\), that at most \(n\) additional purchases are needed to redeem the gift.[2]