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In probability theory, the birthday problem or birthday paradox concerns the probability that, in a set of n randomly chosen people, some pair of them will have the same birthday. In a group of 23 people, the probability of a shared birthday exceeds 50%, while a group of 70 has a 99.9% chance of a shared birthday. (By the pigeonhole principle, the probability reaches 100% when the number of people reaches 367, since there are only 366 possible birthdays, including February 29)
These conclusions are based on the assumption that each day of the year is equally probable for a birthday. Actual birth records show that different numbers of people are born on different days. In this case, it can be shown that the number of people required to reach the 50% threshold is 23 or fewer.
The birthday problem is a veridical paradox: a proposition that at first appears counterintuitive, but is in fact true. While it may seem surprising that only 23 individuals are required to reach a 50% probability of a shared birthday, this result is made more intuitive by considering that the comparisons of birthdays will be made between every possible pair of individuals. With 23 individuals, there are (23 × 22) / 2 = 253 pairs to consider, which is well over half the number of days in a year (182.5 or 183).
Real-world applications for the birthday problem include a cryptographic attack called the birthday attack, which uses this probabilistic model to reduce the complexity of finding a collision for a hash function, as well as calculating the approximate risk of a hash collision existing within the hashes of a given size of population.
The history of the problem is obscure. The result has been attributed to Harold Davenport; however, a version of what is considered today to be the birthday problem was proposed earlier by Richard von Mises.