Birthday problem
In probability theory, the birthday problem or birthday paradox concerns the probability that, in a set of 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. However, 99.9% probability is reached with just 70 people, and 50% probability with 23 people. 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. For example, if half the people were born on one day and the other half on another day, then any two people would have a 50% chance of sharing a birthday.
It may well seem surprising that a group of just 23 individuals is required to reach a probability of 50% that at least two individuals in the group have the same birthday: this result is perhaps made more plausible by considering that the comparisons of birthday will actually be made between every possible pair of individuals = 23 × 22/2 = 253 comparisons, which is well over half the number of days in a year, as opposed to fixing on one individual and comparing his or her birthday to everyone else's. The birthday problem is not a "paradox" in the literal logical sense of being self-contradictory, but is merely unintuitive at first glance.
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.
Calculating the probability
The problem is to compute an approximate probability that in a group of 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 365 possible birthdays are equally likely.The goal is to compute, the probability that at least two people in the room have the same birthday. However, it is simpler to calculate, the probability that no two people in the room have the same birthday. Then, because and are the only two possibilities and are also mutually exclusive,
In deference to widely published solutions concluding that 23 is the minimum number of people necessary to have a that is greater than 50%, the following calculation of will use 23 people as an example. 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 22. 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 conjunction of events may be computed using conditional probability: the probability of Event 2 is 364/365, as person 2 may have any birthday other than the birthday of person 1. Similarly, the probability of Event 3 given that Event 2 occurred is 363/365, as person 3 may have any of the birthdays not already taken by persons 1 and 2. This continues until finally the probability of Event 23 given that all preceding events occurred is 343/365. Finally, the principle of conditional probability implies that is equal to the product of these individual probabilities:
The terms of equation can be collected to arrive at:
Evaluating equation gives
Therefore, .
This process can be generalized to a group of people, where is the probability of at least two of the people sharing a birthday. It is easier to first calculate the probability that all birthdays are different. According to the pigeonhole principle, is zero when. When :
where is the factorial operator, is the binomial coefficient and denotes permutation.
The equation expresses the fact that the first person has no one to share a birthday, the second person cannot have the same birthday as the first, the third cannot have the same birthday as either of the first two, and in general the th birthday cannot be the same as any of the preceding birthdays.
The event of at least two of the persons having the same birthday is complementary to all birthdays being different. Therefore, its probability is
The following table shows the probability for some other values of :
Leap years. If we substitute 366 for 365 in the formula for, a similar calculation shows that for leap years, the number of people required for the probability of a match to be more than 50% is also 23; the probability of a match in this case is 50.6%.
Approximations
The Taylor series expansion of the exponential functionprovides a first-order approximation for for :
To apply this approximation to the first expression derived for, set. Thus,
Then, replace with non-negative integers for each term in the formula of until, for example, when,
The first expression derived for can be approximated as
Therefore,
An even coarser approximation is given by
which, as the graph illustrates, is still fairly accurate.
According to the approximation, the same approach can be applied to any number of "people" and "days". If rather than 365 days there are, if there are persons, and if, then using the same approach as above we achieve the result that if is the probability that at least two out of people share the same birthday from a set of available days, then:
A simple exponentiation
The probability of any two people not having the same birthday is. In a room containing n people, there are pairs of people, i.e. events. 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. In short can be multiplied by itself times, which gives usSince this is the probability of no one having the same birthday, then the probability of someone sharing a birthday is
Poisson approximation
Applying the Poisson approximation for the binomial on the group of 23 people,so
The result is over 50% as previous descriptions. This approximation is the same as the one above based on the Taylor expansion that uses.
Square approximation
A good rule of thumb which can be used for mental calculation is the relationwhich can also be written as
which works well for probabilities less than or equal to. In these equations, is the number of days in a year.
For instance, to estimate the number of people required for a chance of a shared birthday, we get
Which is not too far from the correct answer of 23.
Approximation of number of people
This can also be approximated using the following formula for the number of people necessary to have at least a chance of matching:This is a result of the good approximation that an event with probability will have a chance of occurring at least once if it is repeated times.
Probability table
The lighter fields in this table show the number of hashes needed to achieve the given probability of collision given a hash space of a certain size in bits. 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, or the probability of collision.For comparison, to is the uncorrectable bit error rate of a typical hard disk. In theory, 128-bit hash functions, such as MD5, should stay within that range until about documents, even if its possible outputs are many more.
An upper bound on the probability and a lower bound on the number of people
The argument below is adapted from an argument of Paul Halmos.As stated above, the probability that no two birthdays coincide is
As in earlier paragraphs, interest lies in the smallest such that ; or equivalently, the smallest such that.
Using the inequality in the above expression we replace with. This yields
Therefore, the expression above is not only an approximation, but also an upper bound of. The inequality
implies. Solving for gives
Now, is approximately 505.997, which is barely below 506, the value of attained when. Therefore, 23 people suffice. Incidentally, solving for n gives the approximate formula of Frank H. 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 is 22 or less could also work.
Generalizations
The generalized birthday problem
Given a year with days, the generalized birthday problem asks for the minimal number such that, in a set of randomly chosen people, the probability of a birthday coincidence is at least 50%. In other words, is the minimal integer such thatThe classical birthday problem thus corresponds to determining. The first 99 values of are given here :
A similar calculation shows that = 23 when is in the range 341–372.
A number of bounds and formulas for have been published.
For any, the number satisfies
These bounds are optimal in the sense that the sequence
gets arbitrarily close to
while it has
as its maximum, taken for.
The bounds are sufficiently tight to give the exact value of in 99% of all cases, for example. In general, it follows from these bounds that always equals either
where denotes the ceiling function.
The formula
holds for 73% of all integers. The formula
holds for almost all, i.e., for a set of integers with asymptotic density 1.
The formula
holds for all, but it is conjectured that there are infinitely many counterexamples to this formula.
The formula
holds for all, and it is conjectured that this formula holds for all.
More than 2 people
It is possible to extend the problem to ask how many people in a group are necessary for there to be a greater than 50% probability that at least 3/4/5/etc. of the group share the same birthday.The first few values are as follows: >50% probability of 3 people sharing a birthday - 88 people; >50% probability of 4 people sharing a birthday - 187 people. The full list can be found as sequence A014088 of the Online Encyclopedia of Integer Sequences.
Cast as a collision problem
The birthday problem can be generalized as follows:The generic results can be derived using the same arguments given above.
Conversely, if denotes the number of random integers drawn from to obtain a probability that at least two numbers are the same, then
The birthday problem in this more generic sense applies to hash functions: the expected number of -bit hashes that can be generated before getting a collision is not, but rather only. 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 under the name of capture-recapture statistics to estimate the size of fish population in lakes.
Generalization to multiple types
The basic problem considers all trials to be of one "type". The birthday problem has been generalized to consider an arbitrary number of types. In the simplest extension there are two types of people, say men and women, and the problem becomes characterizing the probability of a shared birthday between at least one man and one woman. The probability of no shared birthdays here iswhere and are Stirling numbers of the second kind. Consequently, the desired probability is.
This variation of the birthday problem is interesting because there is not a unique solution for the total number of people. For example, the usual 50% probability value is realized for both a 32-member group of 16 men and 16 women and a 49-member group of 43 women and 6 men.
Other birthday problems
First match
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? That is, for what is maximum? The answer is 20—if there is a prize for first match, the best position in line is 20th.Same birthday as you
In the birthday problem, neither of the two people is chosen in advance. By contrast, the probability that someone in a room of other people has the same birthday as a particular person is given byand for general by
In the standard case of, substituting gives about 6.1%, which is less than 1 chance in 16. For a greater than 50% chance that one person in a roomful of people has the same birthday as you, would need to be at least 253. This number is significantly higher than : the reason is that it is likely that there are some birthday matches among the other people in the room.
It is not a coincidence that ; a similar approximate pattern can be found using a number of possibilities different from 365, or a target probability different from 50%.
Near matches
Another generalization is to ask for the probability of finding at least one pair in a group of people with birthdays within calendar days of each other, if there are equally likely birthdays.The number of people required so that the probability that some pair will have a birthday separated by days or fewer will be higher than 50% is given in the following table:
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.
Collision counting
The probability that the th integer randomly chosen from will repeat at least one previous choice equals above. The expected total number of times a selection will repeat a previous selection as such integers are chosen equalsAverage number of people
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, 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. It may be shown that if one samples uniformly, with replacement, from a population of size, the number of trials required for the first repeated sampling of some individual has expected value, whereThe function
has been studied by Srinivasa Ramanujan and has asymptotic expansion:
With days in a year, the average number of people required to find a pair with the same birthday is, somewhat more than 23, the number required for a 50% chance. In the best case, two people will suffice; at worst, the maximum possible number of people is needed; but on average, only 25 people are required
An analysis using indicator random variables can provide a simpler but approximate analysis of this problem. For each pair for k people in a room, we define the indicator random variable Xij, for, by
Let X be a random variable counting the pairs of individuals with the same birthday.
For, if, the expected number of with the same birthday is Therefore we can expect at least one matching pair with at least 28 people.
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 2017, in which Paul Keating, the 24th prime minister, and Edmund Barton, the first prime minister, share the same birthday, 18 January.
In the 2014 FIFA World Cup, each of the 32 squads had 23 players. 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. Further results were that psychology students and women did better on the task than casino visitors/personnel or men, but were less confident about their estimates.
Reverse problem
The reverse problem is to find, for a fixed probability,the greatest for which the probability is smaller than the given, or the smallest for which the probability is greater than the given.
Taking the above formula for, one has
The following table gives some sample calculations.
Some values falling outside the bounds have been colored to show that the approximation is not always exact.
Partition problem
A related problem is the partition problem, a variant of the knapsack problem from operations research. Some weights are put on a balance scale; each weight is an integer number of grams randomly chosen between one gram and one million grams. The question is whether one can usually transfer the weights between the left and right arms to balance the scale. If there are only two or three weights, the answer is very clearly no; although there are some combinations which work, the majority of randomly selected combinations of three weights do not. If there are very many weights, the answer is clearly yes. The question is, how many are just sufficient? That is, what is the number of weights such that it is equally likely for it to be possible to balance them as it is to be impossible?Often, people's intuition is that the answer is above. Most people's intuition is that it is in the thousands or tens of thousands, while others feel it should at least be in the hundreds. The correct answer is 23.
The reason is that the correct comparison is to the number of partitions of the weights into left and right. There are different partitions for weights, and the left sum minus the right sum can be thought of as a new random quantity for each partition. The distribution of the sum of weights is approximately Gaussian, with a peak at and width, so that when is approximately equal to the transition occurs. 223 − 1 is about 4 million, while the width of the distribution is only 5 million.