Chapter 5, Problem 4PS

In the carnival game Under-or -Over-Seven, a pair of fair dice is rolled once, and the resulting sum determines whether the player wins or loses his or her bet. For example, the player can bet $\text{\$}1$ that the sum will be under 7 – that is, 2, 3, 4, 5, or 6. For this bet, the player wins $\text{\$}1$ if the outcome equals or is greater than 7. Similarly, the player can bet $\text{\$}1$ that the sum will be over 7– that is, 8, 9, 10, 11 or 12. Here , the player wins $\text{\$}1$ if the result is over 7 but loses $\text{\$}1$ if the result is 7 or under. A third method of play is to bet $\text{\$}1$ on the outcome 7. For this bet, the player wins $\text{\$4}$ if the result of the roll is 7 and loses $\text{\$}1$ otherwise.

a. Construct the probability distribution representing the different outcomes that are possible for a $\text{\$}1$ bet on under 7.

b. Construct the probability distribution representing the different outcome that are possible for $\text{\$}1$ on over 7.

c. Construct the probability distribution representing the different outcome that are possible for $\text{\$}1$ on 7.

d. show that the expected long-run profit (or loss) to the player is the same, no matter which method to play is used.