A393271 Decimal expansion of the probability that at least 3 of 88 randomly selected persons have the same birthday, not considering leap years.

5, 1, 1, 0, 6, 5, 1, 1, 0, 6, 2, 4, 7, 3, 0, 5, 1, 2, 9, 2, 8, 1, 9, 8, 5, 0, 5, 7, 6, 9, 8, 8, 9, 7, 1, 3, 5, 1, 0, 2, 4, 9, 2, 8, 5, 7, 4, 5, 2, 6, 3, 8, 8, 9, 8, 1, 8, 4, 8, 5, 4, 6, 8, 4, 6, 5, 8, 9, 8, 0, 6, 0, 0, 2, 8, 9, 1, 0, 0, 9, 8, 2, 7, 4, 5, 1, 2, 9, 8, 7, 4, 9, 8, 2, 9, 7, 3, 5, 0, 2, 9, 0, 2, 2, 7

Offset

0,1

Comments

88 is the least number of persons for which the probability of shared birthday exceeds 1/2. This result is often attributed to the American mathematician Earl Harry McKinney (1929-2019), although it was already published by the American mathematician Charles F. Pinzka (1918-1984) in 1960.

This constant is a rational number: its numerator and denominator both have 207 digits: 1.219726586003659...*10^206/2.38663638085694...*10^206.

The sequence has a period of 1.92989240612262...*10^159.

Links

Table of n, a(n) for n=0..104.

Rick Durrett, Triple birthday matches in the Senate: Lies, damned lies and chatGPT, arXiv:2302.09643 [math.HO], 2023.

E. H. Mckinney, Generalized Birthday Problem, The American Mathematical Monthly, Vol. 73, No. 4 (1966), pp. 385-387.

C. F. Pinzka, Remarks on some problems in the American Mathematical Monthly, The American Mathematical Monthly, Vol. 67, No. 8 (1960), p. 830.

Eric Weisstein's World of Mathematics, Birthday Problem.

Wikipedia, Birthday problem: Generalizations.

Formula

Equals p(88, 365), where p(n, m) = 1 - (m!*n!/m^n) * Sum_{k = 0..floor(n/2)} 1/(2^k*k!*(n - 2*k)!*(m - n + k)!) is the probability for the general case of n persons, with m days. 88 is the least value of n for which p(n, 365) exceeds 1/2 (Pinzka, 1960).

Example

0.511065110624730512928198505769889713510249285745263...

Mathematica

p[n_, m_] := 1 - (m!*n!/m^n) * Sum[1/(2^k*k!*(n - 2*k)!*(m - n + k)!), {k, 0, Floor[n/2]}];
RealDigits[p[88, 365], 10, 120][[1]]

Prog

(PARI) p(n, m) = 1 - (m!*n!/m^n) * sum(k = 0, n\2, 1/(2^k*k!*(n - 2*k)!*(m - n + k)!));
list(len) = digits(floor(p(88, 365)*10^len));

Crossrefs

Cf. A014088, A225852, A333507, A343015, A393268, A393269, A393270.

Sequence in context: A388818 A201292 A348975 * A271343 A086464 A227577

Adjacent sequences: A393268 A393269 A393270 * A393272 A393273 A393274

Keyword

nonn,cons,easy

Author

Amiram Eldar, Feb 08 2026

Status

approved