# The Birthday Paradox

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. By the pigeonhole principle, the probability reaches 100% when the number of people reaches 367 (since there are 366 possible birthdays, including February 29). However, 99% probability is reached with just 57 people, and 50% probability with 23 people. These conclusions include the assumption that each day of the year (except February 29) is equally probable for a birthday.

The mathematics behind this problem led to a well-known cryptographic attack called the birthday attack, which uses this probabilistic model to reduce the complexity of cracking a hash function.

A graph showing the computed probability of at least two people sharing a birthday amongst a certain number of people.

The birthday problem asks whether any of the people in a given group has a birthday matching any of the others — not one in particular.

In the example given earlier, a list of 23 people, comparing the birthday of the first person on the list to the others allows 22 chances for a matching birthday, the second person on the list to the others allows 21 chances for a matching birthday, third person has 20 chances, and so on. Hence total chances are: 22+21+20+….+1 = 253, so comparing every person to all of the others allows 253 distinct chances (combinations): in a group of 23 people there are pairs.

Presuming all birthdays are equally probable, the probability of a given birthday for a person chosen from the entire population at random is 1/365 (ignoring Leap Day, February 29). Although the pairings in a group of 23 people are not statistically equivalent to 253 pairs chosen independently, the birthday paradox becomes less surprising if a group is thought of in terms of the number of possible pairs, rather than as the number of individuals. Via Birthday problem

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