In defining the notion of independence for three events we found (in Definition 3.13) that we had to check four conditions. If there are four events, say E1, E2, E3, E4, then we have to check 11 conditions - six of the form Pr(Ei ∩ Ej) = Pr(El)Pr(EJ), 1 < i < j < 4; four of the form Pr{Et ∩ Ej ∩ Ek) = Pr(Ei)Pr(Ej)Pr(Ek), 1 < i < j < k < 4; and Pr(E1 ∩ E2 ∩ E3 ∩ E4) = Pr(El)Pr(E2)Pr(E3)Pr(E4).
(a) How many conditions need to be checked for the independence of five events?
(b) How many for n events, where n > 2?