What Is the Birthday Paradox?

In any group of 23 people, the odds that two of them have the same birthday is 50 percent. This phenomenon, which holds true in any group of randomly selected people, is called the birthday paradox. When 57 people are in the group, the probability is 99 percent, and the percentage rises only slightly as the size of the group increases, until it reaches 100 percent at 367 people. If two people meet randomly, however, the chance of them having the same birthday is only 0.27 percent.

More facts about the birthday paradox:

  • The birthday paradox is actually used in mathematics to crack hashing algorithms, and it can be used in cryptography.

  • The reason why the birthday paradox works is because of something called the pigeonhole principle, which states that if there are n number of items placed into m number of holes, and n is more than m, at least one hole will have two items in it.

  • The birthday paradox seems so surprising because people don't tend to go around asking the date of others' birthdays. If they did, it would quickly become apparent that shared birthdays are relatively common.

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Post 6

This shouldn't be called a paradox, since it makes perfect sense (mathematically). There is no seeming contradiction to common sense, merely that most people are surprised at the small number of people to be in the room for the chances of a shared birthday to be greater than a half. Perhaps we should call it the birthday problem or phenomenon.

Another interesting question is: how many people are required for the chances of one of them sharing a birthday with you being greater than 50 percent?

The answer is actually 253 -- much larger than 23.

We could extend this question to birthdays being within one day of each other. How many people would we need in the room before the chance of two of them having birthdays within a day of each other is greater than 50 pecent? What about within two days? And so forth. You guessed correctly if you thought not that many people are required.

@anon: Remember that probability describes the likelihood of a given outcome, not the certainty of it occurring. When you came to the conclusion that probability differs in theory and in practice, that's because in theory, we usually don't account for everything, because that would be impractical. We decide which factors dominate, and include them in the model.

In this case, we have taken everything into consideration, and you can trust the numbers fully. Especially the fact that if there are 367 people (worst case scenario), at least one pair will share the same birthday. The other posters have explained that very clearly. It is not stated, but we assume that all days of the year have the same likelihood as to people being born on that day, except for February 29th. The math isn't hard. Look it up on Wikipedia or something. It's just simple counting probability and some combinatorics.

Post 4

Anon, how many days are there in a year? Every four years there are 366. Otherwise, there are 365 days per year. So if you have 366 persons, and they all have a unique birthday, the 367th person must have a birthday that is the same as one of the former 366 persons.

Post 3

It is correct theoretically on a mathematical level, but not on a practical level. Probability is different in theory than in practice. At least that's what high school taught me.

Post 2

It's not nonsense; it's statistical fact. Not birth date, just birthday, as in April 1, for example. The reason that the probability is 100 percent at 367 people is because there are only 365 days in a year or 366 in a leap year, therefore when there are 367 people in a room, two of them must have the same birthday because there are more people than days of the year.

Post 1

it cannot be correct that there is a 100 percent chance that when 367 people are in a room two people will share the same birthday - that is nonsense.

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