# 400 people are in a room. What is the probability of two random people having the same birthday? [duplicate]

There are 400 people in a room. I pick two people at random. What is the probability that they have the same birthday?

I know that there must be two people in the room who share the same birthday through pigeonhole principle.

But if I pick two people at random I am not sure how to calculate the probability.

• Welcome to Math SE. FYI, a related problem is the Birthday problem. – John Omielan Feb 28 at 19:04
• This is easier than the famous birthday problem linked in the other comment. If we assume (for simplicity, and in the absence of any other data) that each of $365$ possible birthdays is equally likely, then the first person's birthday is irrelevant, and the question is just the probability that the second person's birthday is that one. So $1/365$. It's a little trickier if you take Leap Days (like tomorrow) into account. – Toby Bartels Feb 28 at 19:06
• @amWhy Did you read the previous comment? – NCh Feb 29 at 0:57

The way the problem is stated, the number of people in the room is irrelevant. The probability that any two people have the same birthday is approximately $$\frac{1}{365.25}$$ assuming a uniform distribution of birthdays over time.
• If you are going to say $365.25$ rather than $365$ and talk only about days of the year while ignoring years themselves., then why not use the more accurate $365.2425$? If you were to insist on leap days having some influence, then having a guess as to the age and current year will influence our expectation that there is the chance that one of our selected people were actually born on a leap day – JMoravitz Feb 28 at 19:20