Cast as a collision problem Birthday problem

plot of probability of @ least 1 shared birthday between @ least 1 man , 1 woman

the basic problem considers trials of 1 type . birthday problem has been generalized consider arbitrary number of types. in simplest extension there 2 types of people, m men , n women, , problem becomes characterizing probability of shared birthday between @ least 1 man , 1 woman. (shared birthdays between 2 men or 2 women not count.) probability of no shared birthdays here is

p

0


=


1

d

m
+
n





∑

i
=
1


m



∑

j
=
1


n



s

2


(
m
,
i
)

s

2


(
n
,
j
)

∏

k
=
0


i
+
j
−
1


d
−
k


{\displaystyle p_{0}={\frac {1}{d^{m+n}}}\sum _{i=1}^{m}\sum _{j=1}^{n}s_{2}(m,i)s_{2}(n,j)\prod _{k=0}^{i+j-1}d-k}

where d = 365 , s2 stirling numbers of second kind. consequently, desired probability 1 − p0.

this variation of birthday problem interesting because there not unique solution total number of people m + n. example, usual 50% probability value realized both 32-member group of 16 men , 16 women , 49-member group of 43 women , 6 men.