Question

A set of $1000$ cards numbered 1 through $1000$ is randomly distributed among $1000$ people with each receiving one card. Compute the expected number of cards that are given to people whose age matches the number on the card.

Short answer

One of the people expected to receive a card with a number that matches its age.

Step-by-step solution

Given Information

A set of$1000$ cards numbered $1$ through $1000$ is randomly distributed among $1000$ people with each receiving one card.

Explanation

$X_i=\left\{\begin{array}{l}1,\text{if the}{i}^{th}\text{persons card number matches his}/\text{her age}\\ 0,\text{otherwise}\end{array}\right.$

$X_i$is a new variable.

Expected one of $1,000$people to receive the card which matches its age is,

$E\left(X_i\right)=\frac{1}{1000}$

Explanation

The expected number of cards is,

$X=\sum _{i=1}^{1000}{X}_i$

$E\left(X\right)=\sum _{i=1}^{1000}E\left(X_i\right)$

$=\sum _{i=1}^{1000}\frac{1}{1000}$

$=1000*\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$1000$}\right.$

$=1$

$E\left(X\right)=1$

Final Answer

One of the people expected to receive a card,$E\left(X\right)=1$.