Worked Example 4

In a game, \(3\) green balls and \(7\) yellow balls are placed in a bag. A player draws \(4\) balls at random and without replacement. The number of green balls that she draws is denoted by \(G\). Find the probability distribution of \(G\) and show that \(P\left( G\ge 2 \right)=\frac{1}{3}\). Show that \(\mathrm{E}\left( G \right)=\frac{6}{5}\) and find the variance of \(G\).

Alice scores \(5\) points for each green ball that she draws and Bob scores \(2\) points for each yellow ball that he draws.

Alice’s score is denoted by \(A\) and Bob’s score is denoted by \(B\).

Without finding the probability distribution of the number of yellow balls, find the expectation and variance of \(A-B\) if

1. Alice and Bob play two separate independent games to decide their scores,
2. Alice and Bob decide their score in the same game.