Chapter 8.5, Problem 55E

The Hog Game In the hug game, each player states the number of dice that he or she would like to roll. The player then rolls that many dice. If a 1 comes up on any die, the player's score is 0. Otherwise, the player's score is the sum of the numbers rolled. Source: Mathematics Teacher.

(a) Find the expected value of the player's score when the player rolls one die.

(b) Find the expected value of the player's score when the player rolls two dice.

(c) Verify that the expected nonzero score of a single die is 4, so that if a player rolls n dice that do not result in a score of 0, the expected score is 4n.

(d) Verify that if a player rolls n dice, there are 5ⁿ possible ways to get a nonzero score, and 6ⁿ possible ways to roll the dice. Explain why the expected value, E, of the player's score when the player rolls n dice is then

$E=\frac{5^n(4n)}{6^n}.$