Problem 13 - 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

The expected number of cards that are given to people whose age matches the number on the card is 1.

Step 1: Define the Random Variable

Let us define a random variable X_i, where X_i = 1 if the i-th person's age matches the number on the card they receive, and X_i = 0 otherwise.

Step 2: Calculate the Probability

For each person, the probability of their age matching the number on the card is equal to the probability of that person receiving the card with their age on it. Since there are 1000 cards and each person receives one card, this probability is equal to 1/1000, regardless of the person's age.

Step 3: Calculate the Expected Value for One Person

Using the definition of expected value, we can calculate the expected value for the random variable X_i as follows: \(E(X_i) = 1 * P(X_i = 1) + 0 * P(X_i = 0) = 1 * (1/1000) + 0 = 1/1000\)

Step 4: Calculate the Total Expected Value for All People

Since there are 1000 people, each with their own random variable X_i, we can calculate the total expected value by summing the expected values of all these random variables: \(E(X) = E(X_1) + E(X_2) + ... + E(X_{1000}) = 1000 * (1/1000) = 1\) The expected number of cards that are given to people whose age matches the number on the card is 1.