2022 ASRJC J2 MYCT P2 Q4

Two six-sided fair dice are tossed. The difference between the number of dots that appear on the top faces of the two dice is denoted by the random variable \(X\).

1. Tabulate the probability distribution of \(X\). Show that \(\mathrm{E}\left( X \right)=\frac{35}{18}\) and find \(\mathrm{Var}\left( X \right)\) exactly.[4]

The two dice are tossed by Amber and then followed by Johnny. The difference in the number of dots that appear on the top faces of the two dice for Amber and Johnny is denoted by \({{X}_{1}}\) and \({{X}_{2}}\) respectively.

2. Find the probability that \({{X}_{1}}\) exceeds \({{X}_{2}}\) by at least \(4\).[2]
3. Find the probability that when Amber’s difference exceeds Johnny’s difference by at least \(4\), there is no difference in Johnny’s.[3]