5.4 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, II, 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 $] 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.