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.

i. Construct the probability distribution representing the different outcomes that are possible for a $1 bet on under 7.

ii. Construct the probability distribution representing the different outcomes that are possible for a $1 bet on over 7.

iii. Construct the probability distribution representing the different outcomes that are possible for a $1 bet on 7.

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