Q.7.30 - A First Course in Probability

Question

In the generalized match problem, there are n individuals of whom n_i wear hat size $i,\sum _{i=1}^r n_i=n$. There are also n hats, of which h_i are of size $i,\sum _{i=1}^r h_i=n$. If each individual randomly chooses a hat (without replacement), find the expected number who choose a hat that is their size 

Step-by-Step Solution

Answer

The expected number that chooses a hat that is their size is will be $E\left[X\right]=\frac{1}{n}\sum _{i=1}^r h_i{n}_i$.

Step 1: Given information

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

Step 2: Solution

We need to find the new variable,

$X_{i,j}=\left\{\begin{array}{l}1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\text{ if the }j\text{ th individual choose hat size of }i\\ 0\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\text{ if not }\end{array}\right\}$

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

$X=\sum _{i=1}^r \sum _{j=1}^{n_i} X_{i,j}$

The probability of choosing a hat of size $h_i$ is $\frac{h_i}{n}$

Thus the expected value of X will be,

$E\left[X\right]=E\left[\sum _{i=1}^r \sum _{j=1}^{n_j} X_{i,j}\right]$

$=\sum _{i=1}^r \sum _{j=1}^{n_j} E\left[X_{i,j}\right]$

$=\sum _{i=1}^r \sum _{j=1}^{n_i} \frac{h_i}{n}$

$=\frac{1}{n}\sum _{i=1}^r h_i\times n_i$

$=\frac{1}{n}\sum _{i=1}^r h_i{n}_i$

Step 3: Final answer

The expected number that chooses a hat that is their size is will be $E\left[X\right]=\frac{1}{n}\sum _{i=1}^r h_i{n}_i$