An algorithm and tool for computing exact conditional probabilities of configuration frequencies

When researchers want the exact conditional probability of a configuration frequency, the usual route runs through Fisher's familiar formula for two by two contingency tables, or one of the extensions that push that same idea into tables with more dimensions. That approach works, but it carries baggage once the problem grows.

I take a different route here. The method rests on a combinatorial argument rather than the traditional formula, and the payoff shows up in how it scales: as the number of variables climbs, the combinatorial version holds up far better than the Fisher-based machinery. It also does not care much about how many categories each variable has. In principle the count of categories drops out of the picture entirely.

The setting for all of this is configural frequency analysis (CFA), where multidimensional contingency tables are the everyday object of study and the exact conditional probability of a given cell is what you actually need to report.