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

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.
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1 Answer

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

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