Birthday problem - Wikipedia, the free encyclopedia
In probability theory, the birthday problem or birthday paradox[1] concerns the probability that, in a set ofn 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, 99.9% probability is reached with just 70 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.
In probability theory, the birthday problem or birthday paradox[1] concerns the probability that, in a set of
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.
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 (in fact the 'second' person also has total 22 chances of matching birthday with the others but his/her chance of matching birthday with the 'first' person, one chance, has already been counted with the first person's 22 chances and shall not be duplicated), 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 (232)=23⋅222=253 distinct possible combinations of pairing.
Presuming all birthdays are equally probable,[4] the probability of a given birthday for a person chosen from the entire population at random is 1/365 (ignoring February 29). Although the number of pairings in a group of 23 people is 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.
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