Problem 56 - A First Course in Probability

Question

How many people are needed so that the probability that at least one of them has the same birthday as you is greater than \(\frac{1}{2} ?\)

Step-by-Step Solution

Answer

At least 23 people are needed so that the probability that at least one of them has the same birthday as you is greater than \(\frac{1}{2}\).

Step 1: Define the complementary event

The complementary event is "None of the people have the same birthday as you." We will find the probability of this event and subtract it from 1 to get the probability of at least one person having the same birthday as you.

Step 2: Assume the birthdays are uniformly distributed

We assume that the birthdays are uniformly distributed throughout the year (ignoring leap years), which means that each of the 365 days of the year, a person has an equal probability (\(\frac{1}{365}\)) of being born on that day.

Step 3: Calculate the probability that the first person does not have the same birthday as you

For the first person, the probability of not having the same birthday as you is \(\frac{364}{365}\), as there are 364 days that are not your birthday.

Step 4: Calculate the probability for additional people

If there are \(n\) people, the probability that none of them have the same birthday as you is the product of the probabilities that each person does not have the same birthday as you. So, the probability is \(\left(\frac{364}{365}\right)^n\).

Step 5: Use the complementary probability

The probability of at least one person having the same birthday as you is the complementary probability, which can be calculated as: \(1 - \left(\frac{364}{365}\right)^n\).

Step 6: Determine the number of people needed

We are asked to find the smallest number of people needed such that the probability of at least one person having the same birthday as you is greater than \(\frac{1}{2}\). So we want to find the smallest \(n\) such that: \[1 - \left(\frac{364}{365}\right)^n > \frac{1}{2}\] Since this inequality is hard to solve directly, we'll use trial and error to find the smallest \(n\). Plugging in different values of \(n\), we find that: For \(n = 22\), \(1 - \left(\frac{364}{365}\right)^{22}\approx 0.475\) For \(n = 23\), \(1 - \left(\frac{364}{365}\right)^{23}\approx 0.507\) Thus, at least 23 people are needed so that the probability that at least one of them has the same birthday as you is greater than \(\frac{1}{2}\).