N1999 P2 Q7

In a game, \(2\) red balls and \(8\) blue balls are placed in a bottle The bottle is shaken and Mary draws \(3\) balls at random and without replacement. The number of red balls that she draws is denoted by \(R\). Find the probability distribution of \(R\) and show that \(\mathrm{P}\left( R\ge 1 \right)=\frac{8}{15}\). Show that the expectation of \(R\) is \(\frac{3}{5}\) and find the variance of \(R\).

Mary scores \(4\) points for each red ball that she draws. The balls are now replaced in the bottle and the bottle is shaken again. John draws \(3\) balls at random and without replacement. He scores \(1\) point for each blue ball that he draws. Mary’s score is denoted by \(M\) and John’s score is denoted by \(J\). Find the expectation and variance of \(M-J\).