Today is the birthday of fellow author Kyle (happy birthday, Kyle!). As we wish him a happy 22nd, I thought this would make a perfect opportunity to introduce our readers to statistics… or rather, a specific statistical problem.

The birthday problem is a statistical problem — it’s not pseudo-science, but it is counter-intuitive at first glance. The question is as follows: if you have a group of randomly selected people, with a random distribution of birthdays, what is the probability that two of them will share a birthday? For somebody who is unfamiliar with this problem (or statistics), you might guess that you would need 366 people to ensure that at least two of them share a birthday. As it turns out, you only need 23 people in order to have a 50% possibility of two people sharing a birthday. Once you have a group of 57 people (or more), the probability that two will share a birthday is over 99%. The following is the graph of probabilities from Wikipedia:

Why is this true? This works because of the pigeonhole principle. The easiest way to think about it is a set of mailboxes. If you start randomly slotting letters into a group of mailboxes, then with each successive letter, it will become more and more likely that a mailbox that already has a letter will be given another letter. Keep in mind that we’re not talking about a specific mailbox, but we’re looking for two (or more) letters in any mailbox. So each mailbox that receives a letter will increase the number of mailboxes with a letter in it, and subsequently increase the odds that a mailbox will end up with two or more letters in it next round.

Going back to the birthday problem, we can use the pigeonhole principle to calculate the probability that all of the birthdays are different (given that we have fewer than 365 people). As an example, let’s say that we had four people in a room. The probability that those four people all had different birthdays would be calculated as follows:

p(4) = 1 x (1 - [1/365]) x (1 - [2/365]) x (1 - [3/365])

because for each successive person, when we calculate the probability that their birthday is unique, there is one fewer day that their birthday can occupy. This could also be calculated using the formula:

p(4) = (365 x 364 x 363 x 362) / (365 x 365 x 365 x 365)

Then, to find the probability that any two people share the same birthday, we subtract the probability that they don’t share the same birthday from one. Of course, you can never be 100% sure that you will have two people who share a birthday until you have 366 people, but chances are pretty good (97%) that you’ll have two people who share a birthday by the time you have 50 people, and that percentage will have risen to 99% by the time you have 57 people.

So the next time somebody asks you, or you wonder to yourself, what the chances are that two people share the same birthday — now you know!

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• Wikipedia.org: birthday problem
• Wikipedia.org: pigeonhole principle