2025 JPJC P2 Q7

In a game, a player selects one ball at random from a bag which contains four balls numbered '\(1\)', '\(1\)', '\(2\)' and '\(3\)'. The player notes the number on the selected ball, and then tosses that number of fair coins. The number of heads obtained is denoted by \(X\) and has a probability distribution as follows:

1. Find the exact values of \(a\) and \(b\).[3]

2. Find \(\mathrm{E}(X)\) and show that \(\mathrm{Var}(X)=\frac{39}{64}\).[2]
3. The player plays \(50\) games. Find the probability that the average number of heads obtained per game exceed \(1\).[2]