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Chapter 7: Q.7.15 (page 353)

In Example 2h,say that i andj,i≠j, form a matched pair if i chooses the hat belonging to j and j chooses the hat belonging to i. Find the expected number of matched pairs.

Short Answer

Expert verified

The expected number of matched pairs isE[X]=12

Step by step solution

01

Given Information 

Given in the question that i≠jform a matched pair if i choose the hat belonging to j and j chooses the that belonging to i.

02

Explanation

Let

Xij={1Ifiandjformamatched pair ifichooses the hat belong tojandjchooses the hat belong to0otherwise.

X=∑i=1N∑j=1,i≠jN,Xij-total number

E[X]=∑i=1N∑j=1,i≠jN,E[Xij]

=(N2)1N(N−1)

We get,

=N(N−1)2N(N−1)

=12.

03

Final Answer

The expected number of matched pairs isE[X]=12

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