Question

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) = 3!/4! for each i. (b) Show that P(Ai ? Aj) = 2!/4!, i ? j. (c) Show that P(Ai ? Aj ? Ak) = 1!/4!, i ? j, i ? k, j ? k. (d) Show that the probability of at least one match is P(A1 ? A2 ? A3 ? A4) = 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(A1 ? A2 ? ··· ? An) = 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?

          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) = 3!/4! for each i.
(b) Show that P(Ai ? Aj) = 2!/4!, i ? j.
(c) Show that P(Ai ? Aj ? Ak) = 1!/4!, i ? j, i ? k, j ? k.
(d) Show that the probability of at least one match is
P(A1 ? A2 ? A3 ? A4) = 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(A1 ? A2 ? ··· ? An)
= 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?

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) = 3!/4! for each i.
(b) Show that P(Ai ? Aj) = 2!/4!, i ? j.
(c) Show that P(Ai ? Aj ? Ak) = 1!/4!, i ? j, i ? k, j ? k.
(d) Show that the probability of at least one match is
P(A1 ? A2 ? A3 ? A4) = 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(A1 ? A2 ? ··· ? An)
= 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?

Added by Kathryn G.

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