Four fair sided dice, each with faces marked 1,2,3,4,5,6, are thrown on 7 occasions.Find the probability that the numbers shown on the four dice add up to 5 on exactly one or two of the 7 occasions.

Answer 1

there are 6^4 possible outcomes with 4 dice

there are 4 ways to get a sum of 5
... 3 dice show one, and the 4th shows 2

f = p(5) = 4 / (6^4) ... n = p(not 5) = 1 - [4 / (6^4)]

this is binomial , 5 or (not 5) over 7 throws

(n + f)^7 = n^7 + 7 n^6 f + 21 n^5 f^2 + ... + 7 n f^6 + f^7

the answer is the sum of the 2nd and 3rd terms