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
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.
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