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 A_i denote a match on the ith draw, i = 1, 2, 3, 4.
(a) Show that P(A_i) = 3!/4! for each i.
(b) Show that P(A_i ∩ A_j) = 2!/4!, i ≠ j.
(c) Show that P(A_i ∩ A_j ∩ A_k) = 1!/4!, i ≠ j, i ≠ k, j ≠ k.
(d) Show that the probability of at least one match is
P(A_1 ∪ A_2 ∪ A_3 ∪ A_4) = 1 - 1/2! + 1/3! - 1/4!.
(e) Extend this exercise so that there are n balls in the urn. Show that the probability of at least one match is
P(A_1 ∪ A_2 ∪ ... ∪ A_n)
= 1 - 1/2! + 1/3! - 1/4! + ... + (-1)^(n+1)/n!
= 1 - (1 - 1/1! + 1/2! - 1/3! + ... + (-1)^n/n!).
(f) What is the limit of this probability as n increases without bound?