The birthday problem asks for the probability of a shared birthday in a group of people. In the standard model, the birthdays are independent and each follows the discrete uniform distribution over possible days, where and are positive integers. Let be the probability that no two birthdays match. For , start with an arbitrary person's birthday, then note that the probability that the second person's birthday is different is , that the probability of no match among the first three people is , and so on, up through the th person. The products are interpreted as 1 when . Explicitly,

This can be written in terms of factorials as

For , . For , the probability that two or more people out of a group of do have the same birthday is therefore

while for .

For , let denote the probability that the largest group sharing a birthday has exactly people. Then, for a positive integer , the probability that a birthday is shared by or more people is given by

For , can be computed using the recurrence relation

(Finch 1997), where is the floor function. For the recurrence, use and for , including . For , set as an auxiliary recurrence convention, and use for the weight of an empty assignment.

If 365-day years have been assumed, i.e., the existence of leap days is ignored, and the birthdays are assumed to follow a discrete uniform distribution throughout the year (in actuality, there is a more than 6% increase from the average in September in the United States; Peterson 1998), then the number of people needed for there to be at least a 50% chance that at least two share birthdays is the smallest such that . This is given by , since

The minimum numbers of people needed to obtain for , 2, ..., are 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, ... (OEIS A033810). For and , 2, ..., the minimum numbers of people needed for at least a 50% probability that at least people share a birthday are 1, 23, 88, 187, 313, 460, 623, 798, 985, 1181, 1385, 1596, 1813, ... (OEIS A014088; Diaconis and Mosteller 1989).

For , the probability can be estimated as

where the absolute error of the latter approximation satisfies

(Sayrafiezadeh 1994).

An explicit expression for is

where is a binomial coefficient and is a hypergeometric function. The finite sum holds for all positive and , while the ordinary hypergeometric function expression holds for . This gives the explicit formula for as

where is a regularized hypergeometric function. This expression holds for all positive and , interpreting it as the terminating series with reciprocal gamma function factors when is nonpositive.

For fixed , large , and , an approximation to the number of people needed for to reach is obtained by solving the equation

for the positive solution and taking , where is the ceiling function (Diaconis and Mosteller 1989). For , , and , 3, ..., with the trivial value prepended, this formula gives , 23, 88, 187, 313, 459, 622, 797, 983, 1179, 1382, 1592, 1809, ... (OEIS A050255), whose displayed terms differ from the true values by from 0 to 4. A much simpler but also poorer approximation for such that for and is given by

(Diaconis and Mosteller 1989), whose floor function gives 86, 185, 307, 448, 606, 778, 965, 1164, 1376, 1599, 1832, ... for , 4, ... (OEIS A050256, with its term omitted).

On a cyclic calendar, the "almost" birthday problem, which asks the number of people needed such that two have a birthday within a day of each other, was considered by Abramson and Moser (1970), who showed that 14 people suffice. An approximation for the minimum number of people needed to get a 50-50 chance that two have a match within days out of possible is given by

(Sevast'yanov 1972, Diaconis and Mosteller 1989).