2019 VJC P2 Q8

A bag contains two balls numbered \(3\), \(n\) balls numbered \(2\) and three balls numbered \(1\). A player picks two balls at random from the bag at the same time.

If the difference between the numbers on the two balls is \(2\), the player receives \(\$6\).

If the difference between the numbers on the two balls is \(1\), the player does not receive or lose any money.

If the number on the \(2\) balls are the same, the player loses \(\$1\).

1. Show that the largest value of \(n\) such that player is expected to receive money from this game is \(8\).

For the rest of this question, take the value of \(n\) to be \(8\).

2. Show that the probability that a payer loses money in a game is \(\frac{16}{39}\).