Suppose you are in a room with 33 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 paradox is that the probability is much higher than most people intuitively expect.

Here's how to calculate it:

1. Assume that birthdays are evenly distributed throughout the year, so each person has a 1/365 chance of being born on any particular day.

2. Start by calculating the probability that everyone has a different birthday. For the first person, there are 365 possible days, so the probability that the next person doesn't share a birthday is 364/365. For the third person, the probability is 363/365, and so on.

3. Multiply these probabilities together to get the overall probability that all 33 people have different birthdays. This is calculated as (365/365) * (364/365) * (363/365) * ... * (333/365).

4. The probability that at least two people share a birthday is then 1 minus this number.

5. To get the answer in percentage form, multiply by 100.

This calculation gives a result of approximately 76.040%. So, surprisingly, in a room of just 33 people, there's a 76.040% chance that at least two people share a birthday.