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, using method one, the player can bet $2.00 that the sum will be under 7, that is, 2, 3, 4, 5, or 6. For this bet, the player wins $2.00if the result is under 7 and loses $2.00if the outcome equals or is greater than 7. Similarly, using method two, the player can bet $2.00 that the sum will be over 7, that is, 8, 9, 10, 11, or 12. Here, the player wins $2.00 if the result is over 7 but loses $2.00 if the result is 7 or under. A third method of play is to bet $2.00 on the outcome 7. For this bet, the player wins $8.00 if the result of the roll is 7 and loses $2.00 otherwise. Outcomes of a two dice roll12 3 4 561223343 4 556745675 6 7 8 98456 78910567 8910 11678 910 11 12 a. Construct the probability distribution representing the different outcomes that are possible for a $2.00 bet using method one.XP(X)$.(Type an exact answer in simplified form.)b. Construct the probability distribution representing the different outcomes that are possible for a $2.00 bet using method two.XP(X)$(Type an exact are in simplified)c. Construct the probability distribution representing the different outcomes that are possible for a $2.00 bet using method three.XP(X)$$(Type an exact answer in simplified form.)d. What is the expected long-run profit (or loss) to the player for each of the three methods of play?Method one expected profit (or loss)Method two expected profit (or loss)Method three expected profit (or loss)H = $μ = $μ = $(Round to the nearest cent as needed.)

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College Algebra

10th Edition

ISBN:9781337282291

Author:Ron Larson

Publisher:Ron Larson

Chapter8: Sequences, Series,and Probability

Section8.7: Probability

Problem 4ECP: Show that the probability of drawing a club at random from a standard deck of 52 playing cards is...

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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, using method one, the player can bet $2.00 that the sum will be under 7, that is, 2, 3, 4, 5, or 6. For this bet, the player wins $2.00if the result is under 7 and loses $2.00if the outcome equals or is greater than 7. Similarly, using method two, the player can bet $2.00 that the sum will be over 7, that is, 8, 9, 10, 11, or 12. Here, the player wins $2.00 if the result is over 7 but loses $2.00 if the result is 7 or under. A third method of play is to bet $2.00 on the outcome 7. For this bet, the player wins $8.00 if the result of the roll is 7 and loses $2.00 otherwise.

Outcomes of a two dice roll
1
2 3 4 5
6
1
2
2
3
3
4
3 4 5
567
4567
5 6 7 8 9
8
4
5
6 7
8
9
10
5
67 8
9
10 11
6
7
8 9
10 11 12

a. Construct the probability distribution representing the different outcomes that are possible for a $2.00 bet using method one.
X
P(X)
$.
(Type an exact answer in simplified form.)
b. Construct the probability distribution representing the different outcomes that are possible for a $2.00 bet using method two.
X
P(X)
$
(Type an exact are in simplified)
c. Construct the probability distribution representing the different outcomes that are possible for a $2.00 bet using method three.
X
P(X)
$
$
(Type an exact answer in simplified form.)
d. What is the expected long-run profit (or loss) to the player for each of the three methods of play?
Method one expected profit (or loss)
Method two expected profit (or loss)
Method three expected profit (or loss)
H = $
μ = $
μ = $
(Round to the nearest cent as needed.)

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