2011 SRJC P2 Q6

Two cards are to be drawn at random without replacement from a pack of nine cards numbered \(3\), \(3\), \(3\), \(4\), \(4\), \(4\), \(4\), \(5\), \(5\). The random variable \(Y\) is defined by \(Y={{X}_{1}}-{{X}_{2}}\), where \({{X}_{1}}\) and \({{X}_{2}}\) denote the scores on the first and second card respectively.

1. Show that \(\mathrm{P}\left( Y=1 \right)=\frac{5}{18}\), and tabulate the probability distribution of \(Y\).[5]
2. Find \(\mathrm{P}\left( \left| Y-\mu \right|>\sigma \right)\), where \(\mu \) is the mean of \(Y\) and \(\sigma \) is the standard deviation of \(Y\).[3]
3. Three independent observations of \(Y\) are taken. Find the probability that one of them is positive, one is negative and one is zero.[3]
4. A random sample of \(80\) independent observations of \(Y\) is taken. Find an approximate value for the probability that the total sum of these observation exceeds \(15\).[3]