We select at random an integer between 1 and 100 (so that our sample space is

Ω = {1, 2, 3,…, 100}). We consider the events A1, A2, and A3 which consist of all integers divisible by 2, 5, and 7, respectively.

(i) Show that the events A1 and A3 are independent, but the events A2 and A3 are not.

(ii) If the sample space changes to the set Ω′ = {1, 2, 3,…, 140}, then show that A1, A2, and A3 are pairwise independent. Are they completely independent?