Question 3.17: In the match problem of Example 2.31 involving n, n > 1, ......

Introduction to Probability Models [1578871]

In the match problem of Example 2.31 involving n, n > 1, individuals, find the conditional expected number of matches given that the first person did not have a match.

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Let X denote the number of matches, and let $X_{1}$ equal 1 if the first person has a match and 0 otherwise. Then,

E[X]=E[X|X_{1}=0]P\{X_{1}=0\}+E[X|X_{1}=1]P\{X_{1}=1\}

=E[X|X_{1}=0]\frac{n-1}{n}+E[X|X_{1}=1]\frac{1}{n}

But, from Example 2.31

E[X] = 1

Moreover, given that the first person has a match, the expected number of matches is equal to 1 plus the expected number of matches when n − 1 people select among their own n − 1 hats, showing that

E[X|X_{1}=1]=2

Therefore, we obtain the result

E[X|X_{1}=0]={\frac{n-2}{n-1}}

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