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Chapter 7: Q.7.30 (page 365)

In the generalized match problem, there are n individuals of whom ni wear hat size i,∑i=1r ni=n. There are also n hats, of which hi are of sizei,∑i=1r hi=n. If each individual randomly chooses a hat (without replacement), find the expected number who choose a hat that is their size

Short Answer

Expert verified

The expected number that chooses a hat that is their size is will beE[X]=1n∑i=1r hini.

Step by step solution

01

Given information

The available information is. There are n individuals, ni of them wear a hat of size i. Also, there are n hats, of which hi are of size i.

02

Solution

We need to find the new variable,

Xi,j=1    if thejth individual choose hat size ofi0    if not

The number of individuals that choose a hat of there is given by,

X=∑i=1r ∑j=1ni Xi,j

The probability of choosing a hat of size hiis hin

Thus the expected value of X will be,

E[X]=E∑i=1r ∑j=1nj Xi,j

=∑i=1r ∑j=1nj EXi,j

=∑i=1r ∑j=1ni hin

=1n∑i=1r hi×ni

=1n∑i=1r hini

03

Final answer

The expected number that chooses a hat that is their size is will beE[X]=1n∑i=1r hini

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