Editing Birthday problem (section)

====Generalization to multiple types of people====
[[File:2d birthday.png|thumb|Plot of the probability of at least one shared birthday between at least one man and one woman]]
The basic problem considers all trials to be of one "type". The  birthday problem has been generalized to consider an arbitrary number of types.<ref>[[Michael Christopher Wendl|M. C. Wendl]] (2003) ''[https://dx.doi.org/10.1016/S0167-7152(03)00168-8 Collision Probability Between Sets of Random Variables]'', Statistics and Probability Letters '''64'''(3), 249–254.</ref> In the simplest extension there are two types of people, say {{mvar|m}} men and {{mvar|n}} women, and the problem becomes characterizing the probability of a shared birthday between at least one man and one woman. (Shared birthdays between two men or two women do not count.) The probability of no shared birthdays here is

:<math>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</math>

where {{math|''d'' {{=}} 365}} and {{math|''S''<sub>2</sub>}} are [[Stirling numbers of the second kind]]. Consequently, the desired probability is {{math|1 − ''p''<sub>0</sub>}}.

This variation of the birthday problem is interesting because there is not a unique solution for the total number of people {{math|''m'' + ''n''}}. For example, the usual 50% probability value is realized for both a 32-member group of 16 men and 16 women and a 49-member group of 43 women and 6 men.