Same birthday as you Birthday problem

comparing p(n) = probability of birthday match q(n) = probability of matching birthday

note in birthday problem, neither of 2 people chosen in advance. way of contrast, probability q(n) in room of n other people has same birthday particular person (for example, you), given by

q
(
n
)
=
1
−


(



365
−
1

365


)


n




{\displaystyle q(n)=1-\left({\frac {365-1}{365}}\right)^{n}}

and general d by

q
(
n
;
d
)
=
1
−


(



d
−
1

d


)


n


.


{\displaystyle q(n;d)=1-\left({\frac {d-1}{d}}\right)^{n}.}

in standard case of d = 365 substituting n = 23 gives 6.1%, less 1 chance in 16. greater 50% chance 1 person in roomful of n people has same birthday you, n need @ least 253. note number higher 365/2 = 182 1/2: reason is there birthday matches among other people in room.

it not coincidence 253 = 23 × (23 − 1)/2; similar approximate pattern can found using number of possibilities different 365, or target probability different 50%.