The risk of misunderstanding conditional probabilities – an example from legal departments

Let’s suppose that five law departments lauded everywhere boast total legal spending in terms of revenue less than the median for their respective industries and each of them uses electronic billing software. In other words, 100 percent of those fiscal wizards receive bills electronically.

What are the odds, however, that if your department uses electronic billing software you will have lower-than-average legal spend? Nothing like 100 percent; indeed, a radically lower probability. That condition (the practice of e-billing) may or may not affect the probability of lower total expenses. We would have to compare total legal spending and revenue from all legal departments that have installed such software.

Obviously, I chose e-billing software simply to make a point: if a high proportion of a group of law departments that follow any particular practice show strong results, the conditional probability of similar results will be much lower if we look at all law departments that follow that practice and see their results in metrics.