Why the classic birthday problem is not the whole story
The classic birthday problem assumes every birthday is equally likely and asks for the chance that any two people in one group share a birthday. That assumption is convenient but often unrealistic. When birthdays are not equally likely—say, because some dates are more common—the chance of a match can go up or down depending on which pairs you are watching [3]. A generalized solution therefore has to specify both the distribution of birthdays and which matches count.
Another common variation asks about matches between two groups rather than within one. For example, if you have two classrooms of different sizes, what is the probability that at least one student in the first classroom shares a birthday with at least one student in the second? This is sometimes called the ditype birthday problem, and it has its own generalized solution [5].
When birthdays are uneven or only certain pairs matter
One line of work tackles non-uniform birthday distributions directly. Using methods from graph theory—specifically bounds related to chromatic polynomials—researchers have shown how to compute match probabilities when birthdays are not equally likely and when you care about specific pairs rather than any pair [3]. The key intuition is that unevenness can either raise or lower the overall chance of a match, depending on whether the common birthdays happen to fall on the pairs you are watching.
This matters for real data. If you are checking whether two datasets share individuals, for instance, the 'birthdays' might be combinations of variables, and those combinations are rarely uniform. Generalized collision models can tell you when the number of observed matches is higher than chance would predict, which is exactly the kind of question the classic birthday problem answers in its simplest form [1].
What happens when the number of possible birthdays grows large
A different kind of generalization looks at limiting behavior as the number of possible classes (birthdays) becomes very large. Using tools from point-process theory—essentially a way to describe random arrivals of new objects—researchers have proven that certain birthday-related processes converge to a Poisson process in the limit [4]. This gives distributional limit theorems for many characteristics of the model, not just the basic match probability.
A related approach studies thinned multi-level point processes, where 'thinning' means randomly removing some points to focus on different completion levels. This yields joint limiting distributions for how many classes reach lower completion levels by a random time [2]. These results are technical, but the practical upshot is that generalized birthday problems can be analyzed with the same asymptotic machinery used for other occupancy and collision problems.
Sources used in this answer
When are there too many collisions? Variants of the birthday problem
The paper discusses statistical and probabilistic approaches to detecting when the number of collisions or near-collisions in datasets exceeds chance, relating these methods to the Birthday Problem and Occupancy Problems.
Asymptotic behavior of thinned multi-level point processes in the generalized birthday problem
The paper proves vague convergence of a thinned multi-level point process to a limiting Poisson process with independent levels in a generalized birthday problem, and derives a joint limiting distribution for the number of classes reaching lower completion levels by a random time.
A generalization of the Birthday problem and the chromatic polynomial
The paper generalizes the birthday problem to non-uniform birthday distributions and to specific matching pairs, using bounds on roots of chromatic polynomials to determine whether non-uniformity increases or decreases the probability of a matching pair.
Limit theorems in the generalized birthday problem
The paper establishes limit theorems for various characteristics of a generalized birthday problem by defining a point process on a metric space of countable copies of the real line and showing vague convergence to a Poisson point process as the number of classes tends to infinity.
Shared Ancestors and the Birthday Problem
Using a demographic model and the ditype birthday problem, the paper estimates that the probability is approximately 19%–31% that two randomly chosen African-Americans born around 1960–1965 share at least one African ancestor who was forcibly transported to North America.