A company manages three different mutual funds. Let \(A_{\mathrm{i}}\) 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:

\[\begin{gathered}
P\left(A_{1}ight)=.55, P\left(A_{2}ight)=.60, P\left(A_{3}ight)=.45, P\left(A_{1} \cup A_{2}ight)=.82 \\
P\left(A_{1} \cup A_{3}ight)=.7525, P\left(A_{2} \cup A_{3}ight)=.78, P\left(A_{2} \cap A_{3} \mid A_{1}ight)=.20 \end{gathered}\]

a. Are events \(A_{1}, A_{2}\), and \(A_{3}\) pairwise independent?

b. Are events \(A_{1}, A_{2}\), and \(A_{3}\) independent?

c. What is the probability that funds 1 and 2 both increase 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?