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?