Q.7.12 - A First Course in Probability

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. 

Step-by-Step Solution

Answer

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

Step 1: Given Information

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

Step 2: 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,

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

Step 3: 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.$

$=1$

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

Step 4: Final Answer

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