These ideas can be pooled together to help students see mathematically conditions when Simpson’s Paradox will and will not occur.

If we define the following notation:

A_m = acceptance rate for men in program A

A_w = acceptance rate for women in program A

F_m = acceptance rate for men in program F

F_w= acceptance rate for women in program F

P_m = proportion of men applying to program A (so 1-P_m is the proportion of men applying to program F)

P_w = proportion of women applying to program A (so 1-P_w is the proportion of women applying to program F)

If we assume that women are accepted at a higher rate then men in each program, this implies A_w > A_m and F_W > F_m.

The question is under what conditions can the overall acceptance rate for men be higher than the overall acceptance rate for women. Asked another way, under certain conditions, is the paradox impossible?

To find these overall acceptance rates, we need the weighted averages:

Men: A_mP_m + F_m(1-P_m)

Women: A_wP_w + F_m(1-P_w)

Scenario 1: P_m = P_w (men and women apply to the program A at the same rate P)

Then the weighted averages become:

Men: A_mP + F_m(1-P)

Women: A_wP + F_m(1-P)

Since A_w > A_m and F_W > F_m the overall rate for women must be at least as big as the overall acceptance rate for men.

Scenario 2: A_w = F_w and A_m = F_m (women and as likely to get into Program A as Program F and similar for men)

Then the weighted averages become:

Men: A_mP_m + A_m(1-P_m) = A_m

Women: A_wP_w + A_w(1-P_w) = A_w

Since A_w > A_m then the overall rate for women must be at least as big as the overall acceptance rate for men.

Some generic pictures illustrating these scenarios: