Question

1. Three fair dice are rolled.

   1. (b) What is the probability that there exists at least one pair of dice such that

      the sum of their two results is 6?
      hint: you may want to use the inclusion-exclusion principle:P(A ∪ B ∪ C) =

      P(A) + P(B) + P(C) − P(A ∩ B) − P(A ∩ C) − P(B ∩ C) + P(A ∩ B ∩ C).

Answer 1

Consider the three dies as a, b, and c.

Let A = sum on the faces of dies a and b when we roll three dies = 6

A = sum on the faces of dies b and c when we roll three dies = 6

A = sum on the faces of dies a and c when we roll three dies = 6

Here we want to find  

Because of independence, we have.

Similarly,

and,

Plug these values in the above formula, we get.