N1993 P2 Q7

Alfred and Bertie play a game, each starting with \(\$100\). Two fair dice are thrown. If the total score is \(5\) or more, then Alfred pays \(\$x\), where \(0 < x\le 8\), to Bertie.

If the total score is 4 or less, then Bertie pays \$\(\left( x+8 \right)\) to Alfred.

1. Show that the expectation of Alfred’s cash after the first game is \$\(\frac{1}{3}\left( 304-2x \right)\). Find the expectation of Alfred’s cash after \(6\) games.

2. Find the value of \(x\) for the game to be fair, i.e. for the expectation of Alfred’s winnings to equal the expectation of Bertie’s winnings.
3. Given that \(x=3\), find the variance of Alfred’s cash after the \({{1}^{\mathrm{st}}}\) game.