In the carnival game Under-or-Over-Seven, a pair of fair dice is rolled once, an

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 $1.50 that the sum will be under 7, that is, 2, 3, 4, 5, or 6. For this bet, the player wins $1.50 if the result is under 7 and loses $1.50 if the outcome equals or is greater than 7. Similarly, using method two, the player can bet $1.50 that the sum will be over 7, that is, 8, 9, 10, 11, or 12. Here, the player wins $1.50 if the result is over 7 but loses $1.50 if the result is 7 or under. A third method of play is to bet $1.50 on the outcome 7. For this bet the player wins $6.00 if the result of the roll is 7 and loses $1.50 otherwise. Complete parts (a) through (d). Construct the probability distribution representing the different outcomes that are possible for a $3.00 bet using method 1. B.) Construct the probability distribution representing the different outcomes that are possible for a $3.00 bet using method 2. C.) Construct the probability distribution representing the different outcomes that are possible for a $3 bet using method two. D.) what is the expected long-run profit or loss to the player for each of the three methods of play?

Explanation / Answer

a)

X: gain amount($) +1.5 -1.5

P(X) 15/36 21/36

Since P(sum of numbers on dice is 2,3,4,5,or 6) = 1/36 + 2/36 + 3/36 + 4/36 + 5/36 =15/36

P(Sum of numbers on dice is >=7) = 1 - 15/36 = 21/36

b)

X: gain amount($) +1.5 -1.5

P(X) 15/36 21/36

Since P(sum of numbers on dice is 8,9,10,11,or 12) = 1/36 + 2/36 + 3/36 + 4/36 + 5/36 =15/36

P(Sum of numbers on dice is <=7) = 1 - 15/36 = 21/36

c)

X: gain amount($) 7 -1.5

P(X) 6/36 30/36

d)

Method 1: E(X) = 1.5*(15/36) + (-1.5)*(21/36) = 22.5/36 + -31.5/36 = -9/36 = -$0.25

Method 2: E(X) = 1.5*(15/36) + (-1.5)*(21/36) = 22.5/36 + -31.5/36 = -9/36 = -$0.25

Method 3: E(X) = 7*(6/36) + -1.5*(30/36) = -3/36 = -0.0833

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