A company manages three different mutual funds. Let A; be the event that the ith mutual fund increases in value on a given day. Probabilities of various events relating to the mutual funds are given as follows:
P(Ai) = .55, P(A2) = .60, P( A3 ) = .45,
P(A1 U A2) = .82, P( A1 U A3) = .7525,
P(A2 U A3) = .78, P( A2 n A3 I Ai) = .20.
a. Are events A1 ,A2, and A3 pairwise independent?
b. Are events A 1 , A2, and A3 independent?
c. What is the probability that funds 1 and 2 both in crease in value, given that fund 3 increases in value? Is this different from the unconditional probability that funds 1 and 2 both increase in value?
d. What is the probability that at least one mutual fund will increase in value on a given day?