Intuitive understanding of probability puzzle

Question

Question: The graduate school at the University of California,Berkeley, is composed of two departments - call them A and B. At department A the acceptance rate is 0.5 for both men and women. At department B, however, the rate is 0.1 for men and 0.2 for women. Overall acceptance rate at Berkeley, both departments combined, is 0.3 for men but only 0.25 for women. What fraction of female applicants choose department A?

My (stupid) answer: I tried to devise a primitive formula: $ a*Pa+b*(1-Pa) $
Where a- Percentage of accepted students in A department, b-Percentage of accepted students in B department, Pa- Percentage of accepted A students in sum of all accepted students.

I checked this formula with several different possibilities an it worked, still it may be wrong, but i don't really understand problem intuitively, i don't see the reason behind why the formula is as 'is', if someone understands the problem intuitively, please tell me intuitive way of solving this and similar problems.

Answer 1

This is a famous example of Simpson's Paradox, which actually did occur at the University of California, Berkeley, in 1973, although not with exactly the numbers provided in the question above. A study of this event at the university was published in 1975.

A side note: if you do a web search for "Simpson's Paradox," you will find several accounts of this particular case, some of which (for example, the one linked here) say that the university was sued for sex discrimination. In reality, there was no lawsuit.

Your formula is not quite correct the way you described it, but you are close to the right idea. The thing that seems to have tripped you up is that acceptance rates are not relative to the number of students who are accepted into a program; rather, they are relative to the number of students who apply. If you change the interpretation of $P_a$ so that it signifies the percentage of people who applied to department A among all applicants to the school, then the formula will be correct. (You would have to apply the formula twice, however, once for male applicants and once for female applicants, in order to reproduce the overall acceptance rates for men and women given in the question, since each application of your formula gives only one overall acceptance rate.)

It is even possible that you described the formula incorrectly and then applied it correctly when putting actual numbers into it, in which case you would have gotten the correct numeric results but might have misunderstood what they were saying.

Answer 2

Let $w_A,w_B,$ be the numbers of women who applied to $A$ and $B$, respectively, and $m_A,m_B$ be the same for men.

Then the number of women accepted at $A$ is $\frac{w_A}{2}$ and the number of women accepted at $B$ is $\frac{w_B}{5}.$ The percentage of women accepted to both is:

$$\frac{\frac{w_A}{2}+\frac{w_B}{5}}{w_A+w_B}=0.25$$

The fraction we want is $q=\frac{w_A}{w_A+w_B}$ and note that $\frac{w_B}{w_A+w_B}=1-\frac{w_A}{w_A+w_B}$. So you want:

$$\frac{1}{2}q + \frac{1}{5}(1-q)=0.25$$

Solve for $q$.

The same for men, you get:

$$\frac{1}{2}r + \frac{1}{10}(1-r)=0.3$$