Suppose you are in a room with 27 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.)

To solve this problem, it's easier to calculate the probability that all 27 people have different birthdays and then subtract that from 1.

Step 1: There are 365 days in a year, so the first person has 365 options for a birthday, which is a 100% chance.

Step 2: The second person must have a different birthday from the first, so they have 364 options out of 365. The probability for this is 364/365.

Step 3: The third person must have a different birthday from the first two people, so they have 363 options out of 365. The probability for this is 363/365.

Step 4: Continue this pattern until you get to the 27th person. The 27th person must have a different birthday from the first 26 people, so they have 339 options out of 365. The probability for this is 339/365.

Step 5: The probability that all 27 people have different birthdays is the product of the individual probabilities. So, you multiply all these probabilities together.

Step 6: Subtract the result from step 5 from 1 to get the probability that at least two people share a birthday.

Let's calculate:

P(all different) = (365/365) * (364/365) * (363/365) * ... * (339/365)

Using a calculator, P(all different) ≈ 0.313

So, P(at least 2 same) = 1 - P(all different) = 1 - 0.313 = 0.687

Therefore, the probability that at least two people share a birthday is approximately 68.7%.