Question

Let A, B, and C be disjoint subsets of the sample space. For each one of the following statements, determine whether it is true or false. Note:"False" means "not guaranteed to be true."

P(A) + P(A^c) + P(B) = P(A U A^c U B)

Answer 1

We know that for three subsets A, B and C ,

P(A U B U C ) = P(A) + P(B) + P(C) - P(A and B) - P(A and C) - P(B and C ) + P( A and B and C )

In our case ,

P(A U Ac U B ) = P(A) + P(Ac) + P(B) - P(A and Ac) - P(A and B) - P(Ac and B ) + P( A and Ac and B )

Since A and Ac are disjoint , P(A and Ac) = 0

Since A and B are disjoint , P(A and B) = 0

and also P( A and Ac and B ) = 0

but P(Ac and B ) may not be zero because Ac and B are not disjoint.

Hence, the given statement, P(A) + P(Ac) + P(B) = P(A U Ac U B) is FALSE