Answer to Question #165059 in Statistics and Probability for Fredrick

Question

Let A and B be two events associated with a statistical experiment, with P(A) = 3 4 , P(B) = 1 2 and P(A ∩ B) = 1 4 . Compute (a) P(A ∪ B) (b) P(A|B) (c) P(B|A) (d) P(A ′ ∩ B ′ ) (e) Are A and B independent events? why? (f) Are A and B mutually exclusive events? why?

Answer 1

1. P(A ∪ B) = P(A) + P(B) - P(A ∩ B) = 1/2 + 3/4 - 1/4 = 1
2. P(A|B) = P(A) - P(A ∩ B) = 3/4 - 1/4 = 2/4 = 1/2
3. P(B|A) = P(B) - P(A ∩ B) = 1/2 - 1/4 = 1/4
4. P(A ′ ∩ B ′) = 1 - P(A ∪ B) = 1 - 1 = 0
5. No, A and B are not independent events, because P(A ∩ B) not equal P(A)*P(B).
6. No, because P(A ∩ B) > 0.