Q.7.15 - A First Course in Probability

Question

In Example $2$h,say that i and$j,i\ne j$ , 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.

Step-by-Step Solution

Answer

 The expected number of matched pairs is$E\left[X\right]=\frac{1}{2}$

Step 1: Given Information

Given in the question that  $i\ne j$form a matched pair if i choose the hat belonging to j and j chooses the that belonging to i.

Step 2: Explanation

Let

$X_{ij}=\{\begin{array}{ll}1& \text{If }\mathrm{i} \text{and }\mathrm{j} \text{form a matched pair if }\mathrm{i} \text{chooses the hat belong to }\mathrm{j} \text{and }\mathrm{j} \text{chooses the hat belong to}\\ 0& \text{otherwise.}\end{array}$

$X=\sum _{i=1}^N\sum _{j=1,i\ne j}^{N,}{X}_{ij}\text{-total number}$

$E\left[X\right]=\sum _{i=1}^N\sum _{j=1,i\ne j}^{N,}E\left[X_{ij}\right]$

$=\left(\begin{array}{c}N\\ 2\end{array}\right)\frac{1}{N(N-1)}$

We get,

$=\frac{N(N-1)}{2N(N-1)}$

$=\frac{1}{2}$.

Step 3: Final Answer

The expected number of matched pairs is$E\left[X\right]=\frac{1}{2}$