If there are 25 people in a room, what is the probability that at least two of them share the same birthday? Assume that birthdays are uniformly distributed throughout the year.
We can solve this problem using the complementary probability method. Instead of calculating the probability that at least two people share a birthday, we'll find the probability that no one shares a birthday and then subtract it from 1.

There are 365 days in a year (ignoring leap years for simplicity). The probability that the first person has a unique birthday is 365/365 = 1. For the second person to have a unique birthday, there must be 364 remaining choices out of 365 possible days:

Probability for the second person = 364/365

For the third person to have a unique birthday, there must be 363 remaining choices out of 365 possible days:

Probability for the third person = 363/365

We continue this process until the 25th person:

Probability for the 25th person = 341/365

Now, we multiply all these probabilities together to get the probability that all 25 people have unique birthdays:

P(unique birthdays) = (365/365) * (364/365) * (363/365) * ... * (341/365)

To calculate the probability that at least two people share a birthday, we subtract the probability of everyone having unique birthdays from 1:

P(at least two sharing a birthday) = 1 - P(unique birthdays)
                                                      ≈ 1 - 0.431
                                                      ≈ 0.569

So, the probability that at least two people in a room of 25 share the same birthday is approximately 56.9%.