Probability of identical birth dates?

Question

Multiple people have told me/confirmed that if you have a room with about 30 people in it, chances are at least two people will have the same birthday.

However, no one has been able to explain it using mathematics and logic.

Can anyone explain?

Answer 1

It only takes 23 people in a room for the chances of two people having the same birthday to be greater that 50%.

Calculate the chances of no matching birthdays and subtract from one.

For n people.

P(No) = (365/365) (364/365) (363/365) . . . [(365 - n + 1)/365]

Multiply this out and you will see it dips below 0.50 when n reaches 23.  This means that 1 - the product will be greater that 0.50 when 23 people are in the room.

The probability of matching birthdays when 30 people are in the room is approximately 70.6%.