How would you explain this graph illustration of Simpson's paradox?

Question

I need your help for understanding WHAT in the graph you find in the following link proves Simpson's Paradox. For those who don't know about Simpson's paradox, it is the inversion of the inequalities written in the link. I know for sure that the fact that those two quadrilaterals are symmetrical with respect to the bisector isn't the reason for SP, but what then? Thank you in advance. http://dropseaofulaula.blogspot.it/2012/12/dimostrazioni-senza-parole-il-paradosso.html

Answer 1

Alice and Bob work at the same job for two years. The job consists of a sequence of projects, on which they can succeed or fail. In the first year Alice succeeds on $a_2$ projects and fails on $a_1$, while Bob succeeds on $A_2$ and fails on $A_1$. In the second year Alice succeeds on $b_2$ and fails on $b_1$, while Bob succeeds on $B_2$ and fails on $B_1$. Since

$$\frac{a_2}{a_1}<\frac{A_2}{A_1}\qquad\text{and}\qquad\frac{b_2}{b_1}<\frac{B_2}{B_1}\;,$$

Bob has a better ratio of successes to failures than Alice in each of the two years. But

$$\frac{a_2+b_2}{a_1+b_1}>\frac{A_2+B_2}{A_1+B_1}\;,$$

so overall Alice has a better ratio of successes to failures.

You can convert it this into a statement about probabilities of success, but even if you don’t, it’s Simpson’s paradox, and the graph shows very clearly how it can arise.