Suppose that A1, . . . ,Ak form a sequence of k independent events. Let B1, . . . , Bk be another sequence of k events such that for each value of j (j = 1, . . . , k), either Bj = Aj or Bj = Acj. Prove that B1, . . . , Bk are also independent events. Use an induction argument based on the number of events Bj for which Bj = Acj.