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A total of 2n people, consisting of n married couples, are
Chapter 4, Problem 17STE(choose chapter or problem)
Problem 17STE
A total of 2n people, consisting of n married couples, are randomly divided into n pairs. Arbitrarily number the women, and let Wi denote the event that woman i is paired with her husband.
(a) Find P(Wi).
(b) For i ≠ j, find P(Wi\Wj).
(c) When n is large, approximate the probability that no wife is paired with her husband.
(d) If each pairing must consist of a man and a woman, what does the problem reduce to?
Questions & Answers
QUESTION:
Problem 17STE
A total of 2n people, consisting of n married couples, are randomly divided into n pairs. Arbitrarily number the women, and let Wi denote the event that woman i is paired with her husband.
(a) Find P(Wi).
(b) For i ≠ j, find P(Wi\Wj).
(c) When n is large, approximate the probability that no wife is paired with her husband.
(d) If each pairing must consist of a man and a woman, what does the problem reduce to?
ANSWER:
Solution:
Step1 of 4:
Given that, a total of 2n people, consisting of n married couples, are randomly divided into n pairs.
a). To find
Let denotes the event that women i is paired with her husband.
For any women i, she can pair with any of remaining 2n - 1 people.
So, .