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?