A hint to get started
Separate profit from cash in a dice game
Alfred and Bertie each start with £100. Two fair dice are thrown. If the total is at least 5, Alfred pays £x to Bertie, where . If the total is at most 4, Bertie pays £ to Alfred. Show that Alfred’s expected cash after the first game is £.
Find his expected cash after six games. Find x for a fair game, defined here by equal expected winnings for Alfred and Bertie.
Given , find the variance of Alfred’s cash after the first game.
[J1993/II/7(modified)]
CJC · Lecture · Example 17
Count sums at most 4 in the complete outcome table.
Define profit first, then add the initial cash.
A fair zero-sum game has zero expected net profit.
For x=3, find both profit moments before using the variance identity.