Recall the simulation program of Section 1.8. By suitably modifying this program, find an approximation for the expected profit per game for a player who participates in each of the following games:

(i) We throw two dice simultaneously. If exactly one die lands on a 6, we win $10, if both dice land on 6, we win $30, and if no 6 appears then we lose $5.

(ii) We throw two dice and we receive the amount in dollars equal to the sum of the two outcomes. The cost of entering this game is $7.

(iii) We throw three dice in succession and the amount we receive, in dollars, is equal to the sum of the first two outcomes minus the third outcome. If the difference is negative, then we have to pay an amount equal to that difference.

The cost of entering this game is $3.5.

(iv) We throw three dice in succession and the amount we receive, in dollars, is equal to X1 + X2 2 − X2 3 , where X1, X2, and X3 are the three dice outcomes. If this sum is negative, then we have to pay an amount equal to that sum.

Decide which of the above games (if any) is a fair game.