An urn contains four balls numbered 1 through 4. The balls are selected one at a time without replacement. A match occurs if the ball numbered m is the mth ball selected. Let the event Ai denote a match on the ith draw; i=1,2,3,4.
(a) Show that P(Ai) = 1/4 for each i.
(b) Show that P(Ai ∩ Aj) = 1/12 for i ≠ j.
(c) Show that P(A1 ∩ A2 ∩ A3 ∩ A4) = 1/24.
(d) Show that the probability of at least one match is P(A1 ∪ A2 ∪ A3 ∪ A4).
(e) Extend this exercise so that there are n balls in the urn. Show that the probability of at least one match is P(A1 ∪ A2 ∪ ... ∪ An).
(f) What is the limit of this probability as n increases without bound?