# Question

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.

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.

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