Worked Example 4
In a game, [latex]3[/latex] green balls and [latex]7[/latex] yellow balls are placed in a bag. A player draws [latex]4[/latex] balls at random and without replacement. The number of green balls that she draws is denoted by [latex]G[/latex]. Find the probability distribution of [latex]G[/latex] and show that [latex]P\left( G\ge 2 \right)=\frac{1}{3}[/latex]. Show that [latex]\text{E}\left( G \right)=\frac{6}{5}[/latex] and find the variance of [latex]G[/latex].
Alice scores [latex]5[/latex] points for each green ball that she draws and Bob scores [latex]2[/latex] points for each yellow ball that he draws.
Alice’s score is denoted by [latex]A[/latex] and Bob’s score is denoted by [latex]B[/latex].
Without finding the probability distribution of the number of yellow balls, find the expectation and variance of [latex]A-B[/latex] if
(a)
Alice and Bob play two separate independent games to decide their scores,
(b)
Alice and Bob decide their score in the same game.
Part I
