2023 NYJC P2 Q6

In a funfair, a vendor sells lucky draw tickets at \(\$1\) per ticket. He accepts payment using \(\$1\) or \(\$2\) notes only.

In the queue for tickets, there are \(m\) people each paying with a single \(\$1\) note and \(n\) people each paying with a single \(\$2\) note.

Each person in the queue wants to buy a single ticket and each arrangement of people in the queue is equally likely to occur.

Initially, the vendor has no money but a large supply of tickets. The vendor will stop selling tickets if he cannot give the required change.

1. In the case of \(m\ge 1\) and \(n=1\), find the probability that the vendor can sell one ticket to each person in the queue.[2]
2. By considering the first three people in the queue in the case of \(m\ge 2\) and \(n=2\),
   1. show that the probability that the vendor is able to sell one ticket to each person in the queue is \(\frac{m-1}{m+1}\).[3]
   2. find the probability that the vendor is unable to sell any ticket to the third person, given that the vendor is unable to sell tickets to the first three people in the queue.[3]