Question

Throw two dice. If the sum of the two dice is 6or more, you win $9.If not, you pay me $30.

Step 1 of 2:

Find the expected value of the proposition. Round your answer to two decimal places. Losses must be expressed as negative values.

Answer 1

X : Value of the proposition

Win : X = $9 if the sum of the two dice is 6 or more

Loss : X = -$30 if the sum of the two dice is less than 6 i.e 5 or less

Probability of the sum of two dice is 5 or less

= Probability of the sum of the dice is 5 + Probability of the sum of the dice is 4 + Probability of the sum of the dice is 3 + Probability of the sum of the dice is 2

Probability of the sum of the dice is 5 :

Total number of possible events  when two dice are throw dice (6 x 6 )= 36

Events favoring sum of 5 : (1,4)(2,3)(3,2)(4,1)

Probability of the sum of the dice is 5 : 4/36

Probability of the sum of the dice is 4 :

Events favoring sum of 4 : (1,3)(2,2)(3,1)

Probability of the sum of the dice is 4 : 3/36

Probability of the sum of the dice is 3 :

Events favoring sum of 3 : (1,2)(2,1)

Probability of the sum of the dice is 3 : 2/36

Probability of the sum of the dice is 2 :

Events favoring sum of 2 : (1,1)

Probability of the sum of the dice is 2 : 1/36

Probability of the sum of two dice is 5 or less

= Probability of the sum of the dice is 5 + Probability of the sum of the dice is 4 + Probability of the sum of the dice is 3 + Probability of the sum of the dice is 2

= 4/36 + 3/36 + 2/36 + 1/36 = 10/36

Probability of sum of the two dice is 6 or more = 1-Probability of the sum of two dice is 5 or less = 1-10/36 = 26/36

i.e

X : Value of the proposition

Win : X = $9 if the sum of the two dice is 6 or more i.e P(X=9) = 26/36

Lose : X = -$30 sum of two dice is 5 or less i.e P(X=-30) = 10/36

x	P(x)
9	26/36
-30	10/36

expected value of the proposition : E(X)

expected value of the proposition = - 11/6 = -1.83

expected value of the proposition = - 1.83