Suppose you are in a room with 24 people. If you ignore leap years, what is the probability at least two of you share a birthday?(Round your answer to the nearest thousandths of a percent where needed.)

This is a classic problem in probability theory, often called the "birthday paradox". The counter-intuitive result is that the probability is quite high, even for relatively small groups of people. Here's how you calculate it:

1. Start by assuming that birthdays are evenly distributed throughout the year. This isn't exactly true, but it's close enough for our purposes.

2. There are 365 possible birthdays.

3. The first person in the room can have any birthday, so the probability that at least two people share a birthday is 0 when there's only one person.

4. When the second person enters the room, there's a 364/365 chance that they do NOT share a birthday with the first person (since there are 364 days that are not the first person's birthday).

5. When the third person enters the room, assuming no one shares a birthday yet, there's a 363/365 chance that they do NOT share a birthday with the first two people.

6. You continue this process for all 24 people. So the probability that NO ONE shares a birthday is (365/365) * (364/365) * (363/365) * ... * (342/365).

7. But we want the probability that at least two people DO share a birthday, which is 1 minus the probability that no one shares a birthday.

8. So you calculate that product, subtract it from 1, and that's your answer.

If you do the math, you'll find that the probability that at least two people share a birthday is approximately 0.507, or 50.7%.