Admitted more often in every department, less often overall

Question

A college has two departments, and you can see admissions broken down by department and by applicant gender.

Physics: 20 of 100 women admitted (20%), and 8 of 50 men admitted (16%).
English: 45 of 50 women admitted (90%), and 85 of 100 men admitted (85%).

Women are admitted at a higher rate in Physics and in English. Yet men have the higher overall admission rate.

Show that men are admitted at a higher overall rate, and give the men's overall rate.

Accepted Answer

Within each department, women win: Physics 20\% to 16\%, English 90\% to 85\%. Now pool each gender across both departments. 	ext{Women overall} = \frac{20+45}{100+50} = \frac{65}{150} \approx 43.3\%. 	ext{Men overall} = \frac{8+85}{50+100} = \frac{93}{150} = 62\%. Men win overall, 62\% to 43.3\%, despite losing in both departments. The driver is where each group applied. Most women applied to Physics (100 of 150), which admits almost nobody, so their overall rate is dragged toward that low number. Most men applied to English (100 of 150), which admits nearly everyone, lifting their overall…

Admitted more often in every department, less often overall

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