Calculating the Birthday Number

The history of the problem is obscure. Rouse Ball indicated without citation that it was first discussed by Harold Davenport. The problem is to compute an approximate probability that in a group of n people at least two have the same birthday. For simplicity, variations in the distribution, such as leap years , twins , seasonal, or weekday variations are disregarded, and it is assumed that all possible birthdays are equally likely. Real-life birthday distributions are not uniform, since not all dates are equally likely, but these irregularities have little effect on the analysis.

The goal is to compute P A , the probability that at least two people in the room have the same birthday. In deference to widely published solutions [ which? If one numbers the 23 people from 1 to 23, the event that all 23 people have different birthdays is the same as the event that person 2 does not have the same birthday as person 1, and that person 3 does not have the same birthday as either person 1 or person 2, and so on, and finally that person 23 does not have the same birthday as any of persons 1 through Let these events respectively be called "Event 2", "Event 3", and so on.

One may also add an "Event 1", corresponding to the event of person 1 having a birthday, which occurs with probability 1. This process can be generalized to a group of n people, where p n is the probability of at least two of the n people sharing a birthday. It is easier to first calculate the probability p n that all n birthdays are different.

The event of at least two of the n persons having the same birthday is complementary to all n birthdays being different.

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Therefore, its probability p n is. The following table shows the probability for some other values of n this table ignores the existence of leap years, as described above, as well as assuming that each birthday is equally likely :. Leap years. The first expression derived for p n can be approximated as. According to the approximation, the same approach can be applied to any number of "people" and "days". The probability of no two people sharing the same birthday can be approximated by assuming that these events are independent and hence by multiplying their probability together.

Calculating the Birthday Number

Since this is the probability of no one having the same birthday, then the probability of someone sharing a birthday is. Applying the Poisson approximation for the binomial on the group of 23 people,. A good rule of thumb which can be used for mental calculation is the relation. In these equations, m is the number of days in a year.

The lighter fields in this table show the number of hashes needed to achieve the given probability of collision column given a hash space of a certain size in bits row. Using the birthday analogy: the "hash space size" resembles the "available days", the "probability of collision" resembles the "probability of shared birthday", and the "required number of hashed elements" resembles the "required number of people in a group".

One could also use this chart to determine the minimum hash size required given upper bounds on the hashes and probability of error , or the probability of collision for fixed number of hashes and probability of error. The argument below is adapted from an argument of Paul Halmos. This yields. Therefore, the expression above is not only an approximation, but also an upper bound of p n.

The inequality.

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Solving for n gives. Now, ln 2 is approximately Therefore, 23 people suffice. Mathis cited above. This derivation only shows that at most 23 people are needed to ensure a birthday match with even chance; it leaves open the possibility that n is 22 or less could also work. In other words, n d is the minimal integer n such that.

The classical birthday problem thus corresponds to determining n The first 99 values of n d are given here:. A number of bounds and formulas for n d have been published. In general, it follows from these bounds that n d always equals either.

The formula. Conversely, if n p ; d denotes the number of random integers drawn from [1, d ] to obtain a probability p that at least two numbers are the same, then. This is exploited by birthday attacks on cryptographic hash functions and is the reason why a small number of collisions in a hash table are, for all practical purposes, inevitable. The theory behind the birthday problem was used by Zoe Schnabel [12] under the name of capture-recapture statistics to estimate the size of fish population in lakes.

The basic problem considers all trials to be of one "type". The birthday problem has been generalized to consider an arbitrary number of types. Shared birthdays between two men or two women do not count. The probability of no shared birthdays here is. A related question is, as people enter a room one at a time, which one is most likely to be the first to have the same birthday as someone already in the room? The answer is 20—if there is a prize for first match, the best position in line is 20th. In the birthday problem, neither of the two people is chosen in advance.

By contrast, the probability q n that someone in a room of n other people has the same birthday as a particular person for example, you is given by. Another generalization is to ask for the probability of finding at least one pair in a group of n people with birthdays within k calendar days of each other, if there are d equally likely birthdays. Thus in a group of just seven random people, it is more likely than not that two of them will have a birthday within a week of each other. The expected total number of times a selection will repeat a previous selection as n such integers are chosen equals [15].

In an alternative formulation of the birthday problem, one asks the average number of people required to find a pair with the same birthday. If we consider the probability function Pr[ n people have at least one shared birthday], this average is determining the mean of the distribution, as opposed to the customary formulation, which asks for the median. The problem is relevant to several hashing algorithms analyzed by Donald Knuth in his book The Art of Computer Programming.

An analysis using indicator random variables can provide a simpler but approximate analysis of this problem. An informal demonstration of the problem can be made from the list of Prime Ministers of Australia , of which there have been 29 as of [update] , in which Paul Keating , the 24th prime minister, and Edmund Barton , the first prime minister, share the same birthday, 18 January. An analysis of the official squad lists suggested that 16 squads had pairs of players sharing birthdays, and of these 5 squads had two pairs: Argentina, France, Iran, South Korea and Switzerland each had two pairs, and Australia, Bosnia and Herzegovina, Brazil, Cameroon, Colombia, Honduras, Netherlands, Nigeria, Russia, Spain and USA each with one pair.

Voracek, Tran and Formann showed that the majority of people markedly overestimate the number of people that is necessary to achieve a given probability of people having the same birthday, and markedly underestimate the probability of people having the same birthday when a specific sample size is given. The reverse problem is to find, for a fixed probability p , the greatest n for which the probability p n is smaller than the given p , or the smallest n for which the probability p n is greater than the given p. Some values falling outside the bounds have been colored to show that the approximation is not always exact.

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Tattoos were not the only creative solution overburdened Americans devised—other newspapers repeated the story of the New Jersey man who solved his memory problems by etching his Social Security number on the upper plate of his dentures. Remembering those crucial eight digits was not the only aspect of the new numbers that had Americans somewhat baffled.

Confusion reigned over how many Social Security numbers an individual needed—a new one for each job? Could family members share a number? But somewhere along the line, things changed, and innocent befuddlement and get-rich schemes began to be replaced by suspicion and paranoia. By the early s, Social Security numbers were so widely used in a variety of transactions that fear over the number becoming a national ID helped fuel the passage of the Privacy Act.

And the rise of identify theft fueled fears not just about what the number might signify or how the government could track you with it, but over any request to use the number as identification. We now regard our Social Security number as one of our most closely regarded secrets even as we know we are continually tracked. And now that SSNs have become de facto national identity numbers, the number itself can be evidence for wild theories.

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Using a mixture of public records and the Social Security Death Index , this group claims Obama has used multiple Social Security numbers over the course of his life and favors a number that, when closely analyzed, should belong to someone born in from Connecticut.