A225852 The number of people required for there to be at least a 50% chance that at least 3 share a birthday in a year with n days.

3, 4, 5, 6, 7, 7, 8, 9, 9, 10, 10, 11, 11, 12, 12, 13, 13, 14, 14, 14, 15, 15, 16, 16, 16, 17, 17, 18, 18, 18, 19, 19, 19, 20, 20, 20, 21, 21, 21, 22, 22, 22, 23, 23, 23, 24, 24, 24, 25, 25, 25, 26, 26, 26, 26, 27, 27, 27, 28, 28, 28, 29, 29, 29, 29, 30, 30

Offset: 1

Author

Kevin L. Schwartz and Christian N. K. Anderson, May 17 2013

Keywords

• nonn

Comments

a(365) = 88.

For n <= 1000, a(n) = 1.436 + 1.812*n^0.654 - 0.817/n^3 provides an estimate accurate to 0.6 units.

Examples

			The probability that out of 87 people 3 share a birthday in a year with 365 days is 0.4994549. The corresponding probability for 88 people is 0.5110651. Therefore a(365)=88.
		

Links

• Christian N. K. Anderson, Table of n, a(n) for n = 1..1000
• Christian N. K. Anderson, Table of n and exact probabilities of a(n)-1 and a(n) for n = 1..1000
• Patrice Le Conte, Coincident Birthdays

Crossrefs

Cf. A014088 (n people on 365 days), A033810 (2 people on n days), A225871 (4 people on n days).

Cf. A088141, A182008, A182009, A182010.

Programs

R

library(gmp);#prob of a maximum of exactly k coincident birthdays is
BigQ<-function(nday,p,k) { #nday=days in a year; p=people
    if(p1,sum(sapply(2:k-1,function(j) BigQ(nday-i,p-k*i,j))),1)
    }
    tot
}
BDaySharedByAtLeast<-function(nday,people,k) {
    if(nday<1 | people