Today is my roommate’s birthday and three of my friends’ birthdays. There are 365 days in a year, so what are the chances that so many birthdays are on the same day? More importantly, what are the chances that I have to spend so much time and money on gifts?
The probability is higher than you’d expect – so high, in fact, it has a special name, the birthday paradox. Informally, the birthday paradox is the surprising probabilistic phenomenon that for a relatively small group of people, the probability that at least two people share the same birthday is surprisingly high. For example, if you have 23 people in your stats class, there’s roughly a 50% chance that at least two people have the same birthday.
One way to think about this result is that, at first, it seems like there are only 22 comparisons to be made – essentially 22 ‘chances’ that one person shares his or her birthday with someone else in the room. This is true for the first person, who we’ll call Bob, but what happens when Bob is done comparing his birthday to everyone else’s? Well, then the second person, who we’ll call Mary, compares her birthday to everyone in the room except for Bob, who she has already compared birthdays with. So Mary has 21 people to compare with. The third person has 20 potential matches, the fourth has 19, and so forth. So what you really have is 22+21+…+1 = 253 comparisons, not 22.
Each individual comparison has only a 1/365 chance of being a match – equivalently, each trial has a 99.726% chance of not being a match. This is probably why we’re so surprised by this paradox – we’re conditioned to think it’s rare to have the same birthday as someone, because when you ask any given person when their birthday is there is only a .274% chance you’ll have the same birthday as him or her. But each time we compare, there is one less possibility – i.e. there is one more birth date that isn’t available, so to speak. Below is the probability that no two people in the room share a birthday. Notice that the first term is 365/365 because the first person has 365 birthdays to choose from without ‘taking’ someone else’s. The second person, however, has one less choice, because the first person’s birthday has already been taken. The third person has one less choice than the second person. This continues until the last person, person number 23, has 22 fewer choices (365-22=343) of birthdays if he is to avoid having the same birthday as another person in the room.
The probability that two people in the room do share a birthday is obviously the complement of this, so it’s equal to 1 – 0.4927 = 50.73%. You can write the expression above with factorials etc., but I just wanted to sort of explain why (though not actually justify in a rigorously mathematical sense) the birthday paradox is a thing. For a more hardcore explanation, you can visit Wolfram Mathworld here.