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 $ 1 that the sum will be under 7— that is, 2, 3, 4, 5, or 6. For this bet, the player wins $ 1 if the result is under 7 and loses $ 1 if the outcome equals or is greater than 7. Similarly, the player can bet $ 1 that the sum will be over 7— that is, 8, 9, 10, 11, or 12. Here, the player wins $ 1 if the result is over 7 but loses $ 1 if the result is 7 or under. A third method of play is to bet $ 1 on the outcome 7. For this bet, the player wins $ 4 if the result of the roll is 7 and loses $ 1 otherwise.
a. Construct the probability distribution representing the different outcomes that are possible for a $ 1 bet on under 7.
b. Construct the probability distribution representing the different outcomes that are possible for a $ 1 bet on over 7.
c. Construct the probability distribution representing the different outcomes that are possible for a $ 1 bet on 7.
d. Show that the expected long- run profit (or loss) to the player is the same, no matter which method of play is used.