## Birthday Problem

Consider the probability that no two people out of a group of will have matching birthdays out of equally possible birthdays. Start with an arbitrary person's birthday, then note that the probability that the second person's birthday is different is , that the third person's birthday is different from the first two is , and so on, up through the th person. Explicitly,

 (1)

But this can be written in terms of Factorials as
 (2)

so the probability that two people out of a group of do have the same birthday is therefore
 (3)

If 365-day years have been assumed, i.e., the existence of leap days is ignored, then the number of people needed for there to be at least a 50% chance that two share birthdays is the smallest such that . This is given by , since

 (4)

The number of people needed to obtain for , 2, ..., are 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, ... (Sloane's A033810).

The probability can be estimated as

 (5) (6)

where the latter has error
 (7)

In general, let denote the probability that a birthday is shared by exactly (and no more) people out of a group of people. Then the probability that a birthday is shared by or more people is given by

 (8)

can be computed explicitly as

 (9)

where is a Binomial Coefficient, is a Gamma Function, and is an Ultraspherical Polynomial. This gives the explicit formula for as

 (10)

cannot be computed in entirely closed form, but a partially reduced form is
 (11)
where
 (12)
and is a Generalized Hypergeometric Function.

In general, can be computed using the Recurrence Relation

 (13)

(Finch). However, the time to compute this recursive function grows exponentially with and so rapidly becomes unwieldy. The minimal number of people to give a 50% probability of having at least coincident birthdays is 1, 23, 88, 187, 313, 460, 623, 798, 985, 1181, 1385, 1596, 1813, ... (Sloane's A014088; Diaconis and Mosteller 1989).

A good approximation to the number of people such that is some given value can be given by solving the equation

 (14)

for and taking , where is the Ceiling Function (Diaconis and Mosteller 1989). For and , 2, 3, ..., this formula gives , 23, 88, 187, 313, 459, 722, 797, 983, 1179, 1382, 1592, 1809, ..., which differ from the true values by from 0 to 4. A much simpler but also poorer approximation for such that for is given by
 (15)

(Diaconis and Mosteller 1989), which gives 86, 185, 307, 448, 606, 778, 965, 1164, 1376, 1599, 1832, ... for , 4, ....

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

 (16)

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

References

Abramson, M. and Moser, W. O. J. More Birthday Surprises.'' Amer. Math. Monthly 77, 856-858, 1970.

Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recreations and Essays, 13th ed. New York: Dover, pp. 45-46, 1987.

Bloom, D. M. A Birthday Problem.'' Amer. Math. Monthly 80, 1141-1142, 1973.

Bogomolny, A. Coincidence.'' http://www.cut-the-knot.com/do_you_know/coincidence.html.

Clevenson, M. L. and Watkins, W. Majorization and the Birthday Inequality.'' Math. Mag. 64, 183-188, 1991.

Diaconis, P. and Mosteller, F. Methods of Studying Coincidences.'' J. Amer. Statist. Assoc. 84, 853-861, 1989.

Feller, W. An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd ed. New York: Wiley, pp. 31-32, 1968.

Finch, S. Puzzle #28 [June 1997]: Coincident Birthdays.'' http://www.mathsoft.com/mathcad/library/puzzle/soln28/soln28.html.

Gehan, E. A. Note on the Birthday Problem.''' Amer. Stat. 22, 28, Apr. 1968.

Heuer, G. A. Estimation in a Certain Probability Problem.'' Amer. Math. Monthly 66, 704-706, 1959.

Hocking, R. L. and Schwertman, N. C. An Extension of the Birthday Problem to Exactly Matches.'' College Math. J. 17, 315-321, 1986.

Hunter, J. A. H. and Madachy, J. S. Mathematical Diversions. New York: Dover, pp. 102-103, 1975.

Klamkin, M. S. and Newman, D. J. Extensions of the Birthday Surprise.'' J. Combin. Th. 3, 279-282, 1967.

Levin, B. A Representation for Multinomial Cumulative Distribution Functions.'' Ann. Statistics 9, 1123-1126, 1981.

McKinney, E. H. Generalized Birthday Problem.'' Amer. Math. Monthly 73, 385-387, 1966.

Mises, R. von. Über Aufteilungs--und Besetzungs-Wahrscheinlichkeiten.'' Revue de la Faculté des Sciences de l'Université d'Istanbul, N. S. 4, 145-163, 1939. Reprinted in Selected Papers of Richard von Mises, Vol. 2 (Ed. P. Frank, S. Goldstein, M. Kac, W. Prager, G. Szegö, and G. Birkhoff). Providence, RI: Amer. Math. Soc., pp. 313-334, 1964.

Riesel, H. Prime Numbers and Computer Methods for Factorization, 2nd ed. Boston, MA: Birkhäuser, pp. 179-180, 1994.

Sayrafiezadeh, M. The Birthday Problem Revisited.'' Math. Mag. 67, 220-223, 1994.

Sevast'yanov, B. A. Poisson Limit Law for a Scheme of Sums of Dependent Random Variables.'' Th. Prob. Appl. 17, 695-699, 1972.

Sloane, N. J. A. A014088 and A033810 in An On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html.

Stewart, I. What a Coincidence!'' Sci. Amer. 278, 95-96, June 1998.

Tesler, L. `Not a Coincidence!'' http://www.nomodes.com/coincidence.html.