B'', otherwise it would be no paradox. The odds(!) are 5/6 in A'' and 3/3 in B'', the probabilities are 5/11 and 3/6. This paradox has the awkward consequence that you can have four studies with the results pointing unanimously in one direction and when you pool the results to get a clearer picture the combined results point in the opposite direction. Scientists just hate such a situation.

To use a real life example: When you are a parent of now adults kids or just watch the young generation growing you are not surprised to learn that the average size (height) of male German students has increased since the 1950s. Similarly, the average size of female students in Germany has increased during the same time. However, if you pool the results to get the average size of all students in Germany, both male and female together, you'll find that the average size has decreased. The problem is most times not as easy to spot as in this example.

There was even a court case in California in which a woman whose application to a Californian university was rejected sued for violation of equal opportunity (or whatever the term is) when she found out that the university did reject more females than males on a percentage base. She lost, because she fell prey to Simpson's paradox, when the university could show that (though the data pooled for the university showed a higher percentage of males being accepted) for each faculty separately there was even a small bias in favour women.

Jurists must hate that: If they want to do justice to equal opportunity on the university level, they necessarily do injustice to males at the faculty level. If they want to do justice on a faculty level, they must do injustice to woman at the whole university level. And all that for purely mathematical reasons that do not bow to justice.

Wolfgang