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)
			(2)

But this can be written in terms of factorials as

(3)

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

			(4)
			(5)

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

(6)

In general,  can be computed using the recurrence relation

(7)

(Finch 1997). However, the time to compute this recursive function grows exponentially with  and so rapidly becomes unwieldy.

If 365-day years have been assumed, i.e., the existence of leap days is ignored, and the distribution of birthdays is assumed to be uniform 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

			(8)
			(9)

The number  of people needed to obtain  for , 2, ..., are 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, ... (OEIS A033810). 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, ... (OEIS A014088; Diaconis and Mosteller 1989).

The probability  can be estimated as

			(10)
			(11)

where the latter has error

(12)

(Sayrafiezadeh 1994).

 can be computed explicitly as

			(13)
			(14)

where  is a binomial coefficient and  is a hypergeometric function. This gives the explicit formula for  as

			(15)
			(16)

where  is a regularized hypergeometric function.

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

(17)

for  and taking , where  is the ceiling function (Diaconis and Mosteller 1989). For  and , 2, 3, ..., this formula gives , 23, 88, 187, 313, 459, 622, 797, 983, 1179, 1382, 1592, 1809, ... (OEIS A050255), 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

(18)

(Diaconis and Mosteller 1989), which gives 86, 185, 307, 448, 606, 778, 965, 1164, 1376, 1599, 1832, ... for , 4, ... (OEIS A050256).

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

(19)

(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.org/do_you_know/coincidence.shtml.

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

Diaconis, P. and Mosteller, F. "Methods for 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.mathcad.com/library/LibraryContent/puzzles/puzzle.asp?num=28.

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.

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

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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.

Peterson, I. "MathTrek: Birthday Surprises." Nov. 21, 1998. http://www.sciencenews.org/sn_arc98/11_21_98/mathland.htm.

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. Sequences A014088, A033810, A050255, and A050256 in "The On-Line Encyclopedia of Integer Sequences."

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