Introduction to Probability, Second Edition, 2nd Edition

Let’s use event notation to make this precise. For events A, B, and C, we say that we have a Simpson’s paradox if

$P\left(A|B,C\right)<P\left(A|B^c,C\right)$

$P\left(A|B,C^c\right)<P\left(A|B^c,C^c\right),$

but

$P\left(A|B\right)>P\left(A|B^c\right).$

In this case, let A be the event of a successful surgery, B be the event that Dr. Nick is the surgeon, and C be the event that the surgery is a heart surgery. The conditions for Simpson’s paradox are fulfilled because the probability of a successful surgery is lower under Dr. Nick than under Dr. Hibbert whether we condition on heart surgery or on Band-Aid removal, but the overall probability of success is higher for Dr. Nick.

The law of total probability tells us mathematically why this can happen:

$\begin{array}{ll}\hfill P\left(A|B\right)& =P\left(A|C,B\right)P\left(C|B\right)+P\left(A|C^c,B\right)P\left(C^c|B\right)\\ \hfill P\left(A|B^c\right)& =P\left(A|C,B^c\right)P\left(C|B^c\right)+P(A|C^c,B^{}\end{array}$