Chapter 7, Problem 7.56P

Consider the following dice game: A pair of dice is rolled. If the sum is 7, then the game ends and you win 0. If the sum is not 7, then you have the option of either stopping the game and receiving an amount equal to that sum or starting over again. For each value of $i,i=\mathrm{2.....}12$, find your expected return if you employ the strategy of stopping the first time that a value at least as large as i appears. What value of i leads to the largest expected return?

Hint: Let $X_i$ denote the return when you use the critical value i. To compute $E\left[X_i\right]$ condition on the initial sum.