A091715 -id:A091715 - cp's OEIS frontend

A091673 Numerator Q of probability P = Q(n)/365^(n-1) that exactly two out of n people share the same birthday.

1, 1092, 793884, 480299820, 261163522620, 132358677731280, 63798093049771080, 29612552769907347240, 13345042642324219106280, 5872442544965392834838400, 2533775368098060137659608000, 1075256447734638237381213700800

Offset: 2

Hugo Pfoertner, Feb 03 2004

A 365-day year and a uniform distribution of birthdays throughout the year are assumed.

			a(3)=1092 because the probability that in a group of 3 people exactly two of them share the same birthday is (1/365^3)*3!*binomial(365,1)*binomial(364,1)/2 = (1/365^2)*3*364 = (1/365^2)*1092.

Patrice Le Conte, Coincident Birthdays.
Eric Weisstein's World of Mathematics, Birthday Problem.

Cf. A014088, A091674 gives probabilities for two or more coincidences, A091715 gives probabilities for three or more coincidences.

P[n_] := (n! Sum[ Binomial[365, i]*Binomial[365 - i, n - 2i] /2^i, {i, 1, Floor[n/2]}]/365); Table[ P[n], {n, 2, 13}] (* Robert G. Wilson v, Feb 09 2004 *)

from math import factorial, comb
from fractions import Fraction
def A091673(n): return int(factorial(n)*sum(Fraction(comb(365,i)*comb(365-i,n-(i<<1)),365<>1)+1))) # Chai Wah Wu, Jan 22 2025

P(n) = n!*Sum_{i=1..floor(n/2)} binomial(365, i)*binomial(365-i, n-2*i)/2^i.

More terms from Robert G. Wilson v, Feb 09 2004

A091674 Numerator Q of probability P = Q(n)/365^(n-1) that two or more out of n people share the same birthday.

Original entry on oeis.org

1, 1093, 795341, 481626601, 262130079485, 132974790903865, 64157156143943045, 29808728817823292065, 13447118719710220490765, 5923562823392985950002825, 2558600264156303883127171925, 1087010123072386037371040127025

Offset: 2

Hugo Pfoertner, Feb 03 2004

A 365-day year and a uniform distribution of birthdays throughout the year are assumed.

Patrice Le Conte, Coincident Birthdays.
Mathforum at Drexel, The Birthday Problem, Ask Dr. Math: FAQ.
Eric Weisstein's World of Mathematics, Birthday Problem.

Cf. A014088, A091673 (probabilities for exactly two), A091715 (probabilities for three or more).

Q[n_] := (1 - Product[(1 - i/365), {i, 1, n - 1}])365^(n - 1); Table[ Q[n], {n, 2, 13}] (* Robert G. Wilson v, Feb 05 2004 *)

from math import prod
def A091674(n): return 365**(n-1)-prod(365-i for i in range(1,n)) # Chai Wah Wu, Jan 22 2025

Q(n) = (1 - Product_{i=1..n-1} (1-i/365))*365^(n-1).

More terms from Robert G. Wilson v, Feb 05 2004

cp's OEIS Frontend

A091673 Numerator Q of probability P = Q(n)/365^(n-1) that exactly two out of n people share the same birthday.

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