What is the probability that there are at least two people with the same birthday in a class of 40 students?

Old, old problem with surprising answers

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To calculate the probability that there are at least two people with the same birthday in a class of 40 students, we need to use the concept of the Birthday Paradox.

The Birthday Paradox states that the probability of two or more people sharing the same birthday becomes surprisingly high even in a small group.

To calculate the probability, we can use the complement rule. The complement rule states that the probability of an event happening is equal to 1 minus the probability of it not happening. In this case, we want to find the probability that no two people have the same birthday.

Let's break down the problem step by step:

1. Calculate the probability that the first person has a unique birthday. Since there are 365 days in a year, the probability is 365/365.

2. Calculate the probability that the second person has a unique birthday. In this case, there are 364 remaining days for the second person's birthday, so the probability is 364/365.

3. Continue this process for each subsequent person until the 40th person. The probability for the 40th person will be 326/365.

4. Calculate the complement of the event, which is the probability that none of the 40 people have the same birthday. To do this, multiply all the individual probabilities together:

(365/365) * (364/365) * (363/365) * ... * (326/365)

5. Finally, subtract this result from 1 to get the probability that at least two people share the same birthday in a class of 40 students.

1 - (365/365) * (364/365) * (363/365) * ... * (326/365)

Calculating this expression will give us the desired probability.