true story about probability? A women's organization was contemplating suing a famous American university when it learned that the percentage of women who received tenure in the university was smaller than the percentage of men. But then it was discovered that in every department, the percentage of women who received tenure in that department was greater than the percentage of men who did. How can that be?
 A: This is called Simpson's paradox.  The way it can happen is that women tended to be a larger proportion of the faculty in departments where fewer people get tenure.  Imagine all but one or two of the women going into a department where one out of $20$ faculty members get tenure, and all but one or two of the the men going into a department that gives tenure to everyone except one man.  That first department could deny tenure to all of the few men, and women would still fare worse in the university as a whole.
A: First, let us take a look at this question.

One year Babe Ruth had a higher batting average than Lou Gehrig for
  the first half of the season and also for the second half of the season. But
  Lou Gehrig had a higher batting average for the entire season. How can
  that be?

The batting average for a baseball player over a time period $T$ is$${{\#\text{ of hits during }T\text{ }(\text{successful batting attempts})}\over{\#\text{ counted at-bats }(\text{batting attempts})}}.$$The essential idea is that the proportion of at-bats in the first half of the season versus the second half of the season may differ for Lou Gehrig and Babe Ruth. While the batting average for the season is a weighted average of the batting averages for the first and second halves, the weights depend on the proportions of counted at-bats in the two halves. The players' proportions must have been different.
It can be proven that the batting averages of both players must have been higher in one half of the season than the batting averages of both players in the other half of the season. $($Conversely, given such a set of averages, there is some set of weights such that the total batting averages are "reversed."$)$ It must be that there were disproportionately more at-bats for Lou Gehrig in the half of the season in which both players performed well.
For example, suppose Babe Ruth hit $1/9$ times in the first half and $1/1$ times in the second while Lou Gehrig hit $0/1$ times in the first and $8/9$ in the second. Then the batting averages are Babe Ruth $($$.111$, $1.000$, $.200$$)$ and Lou Gehrig $($$.000$, $.889$, $.800$$)$ for the time periods $($first, second, total$)$.

Now let us head back to the original problem. It is entirely analogous to the batting average problem above. Let $D_1, D_2, \dots, D_N$ denote the different departments in the university, and$$w_k = \text{total number of women in department }D_k,$$$$w_k' = \text{number of tenured women in }D_k,$$$$m_k = \text{total number of men in }D_k,$$$$m_k' = \text{number of tenured men in }D_k.$$As seen in the solution to the batting average problem, we may have $${{w_k'}\over{w_k}} > {{m_k'}\over{m_k}}$$for all $k \le N$, and yet have$${{\sum_{k=1}^N w_k'}\over{\sum_{k=1}^N w_k}} < {{\sum_{k=1}^Nm_k'}\over{\sum_{k=1}^N m_k}}.$$There must be a department $D_\ell$ where department is relatively uncommon $($so that $w_\ell' < m_k$ for at least one $k \le N$$)$; we must also have$${{w_\ell}\over{\sum_{k=1}^N w_k}} > {{m_\ell}\over{\sum_{k=1}^N m_k}},$$that is, $D_\ell$ must employ a large proportion of the female professors and a relatively small proportion of the male professors.
