Chapter 7, Problem 7.48P

Consider the following dice game. as played at a certain gambling casino: Players 1 and 2 roll a pair of dice in turn. The bank then rolls the dice to determine the outcome according to the following rule: Player $i,i=1,2,$ wins if his roll is strictly greater than the banks. For $i=1,2,$ let $I_i=\{\begin{array}{c}1\text{if }i\text{ wins}\\ 0\text{otherwise}\end{array}$ and show that $I_1$ and $I_2$ are positively correlated. Explain why this result was to be expected.