A pair of dice are rolled 8 times. Find the probability that:

Question

a. A sum of 7 is rolled exactly 2 times.

b. A sum of 7 is rolled at least 2 times.

c. A sum of 7 is rolled at most 2 times.

d. A sum of 7 or 11 is rolled exactly 5 times.

e. A sum of 7 or 11 is never rolled.

f. A sum of 7 or 11 is rolled the 3rd and 5th time only.

Answer 1

a) P(sum = 7) = 6/36 = 1/6

P(2 times of sum = 7) = 8C2 (1/6)^2 ( 5/6)^6 = .2605

 

b) P(at least 2 times of sum = 7) = 1 - ∑(i=0, 1) 8Ci (1/6)^i ( 5/6)^(8-i) = .3953

 

c) P(at most 2 times of sum = 7) = ∑(i=0, 2) 8Ci (1/6)^i ( 5/6)^(8-i) = .8652

 

d) P(sum = 7 or 11) = 8/36 = 2/9

P(A sum of 7 or 11 is rolled exactly 5 times) = 8C5 (2/9)^5 ( 7/9)^3 = .0143

 

 

e) P(A sum of 7 or 11 is never rolled) = ( 7/9)^8= .1339

 

f) P(A sum of 7 or 11 is rolled the 3rd and 5th time only) = (2/9)^2 (7/9)^6 = .0109