2023 SAJC Tutorial 20 Q1

Three cards are to be drawn at random without replacement from a pack of six cards numbered \(0\), \(0\), \(1\), \(1\), \(1\), \(1\).

The random variables \({{X}_{1}}\), \({{X}_{2}}\) and \({{X}_{3}}\) denotes the numbers on the first, second and third card respectively.

1. If \(Y={{X}_{1}}-{{X}_{2}}+{{X}_{3}}\), show that \(P\left( Y=1 \right)=\frac{1}{3}\) and find the probability distribution of \(Y\).[3]
2. Find the values of \(\mathrm{E}\left( Y \right)\) and \(\mathrm{Var}\left( Y \right)\).[2]