Probability of sharing a birthday

[locke]

I'm trying to figure out what the probability of two people in a room sharing a birthday is, if there are 19 people in the room.

Originally i assumed that the chances of any pair of people sharing a birthday are 1/365. Since there are 19c2 pairs of people in the room in question, I thought the probability was 19c2/365. This is obviously wrong though, since if the number of pairs in the room was >365, this type of reasoning would yield probabilities greater than one.

Can anyone point me towards the right answer?

 

[ceptimus]

Since you are concerned with whether two or more people share the same birthday, it's easier to calculate the probablilty that a birthday is not shared, and then subtract that from 1.

The probability that 4 people will all have different birthdays (ignoring leap years) is:

364/365 * 363/365 * 362/365

Explanation: We don't worry about the first person, as a single person can't have a shared birthday. The second person has 364 days to choose from, the third person 363, and the fourth person 362.

Extending this to 19 people, we get the probability that they all have different birthdays is:

(364! / (364 - (19 - 1))!) / 365^(19 - 1)

So the probability that two (or more) people share a birthday is one minus that.

1 - (364! / 346!) / 365^18 = 0.379119 (approximately)