2017 NJC P2 Q7

There are three identically shaped balls, numbered from \(1\) to \(3\), in a bag. Balls are drawn one by one at random and with replacement. The random variable \(X\) is the number of draws needed for any ball to be drawn a second time. The two draws of the same ball do not need to be consecutive.

1. Show that \(\mathrm{P}\left( X=4 \right)=\frac{2}{9}\) and find the probability distribution of \(X\).[3]
2. Show that \(\mathrm{E}\left( X \right)=\frac{26}{9}\) and find the exact value of \(\mathrm{Var}\left( X \right)\).[3]
3. The mean for forty-four independent observations of \(X\) is denoted by \(\bar{X}\) . Using a suitable approximation, find the probability that \(\bar{X}\) exceeds \(3\).[3]