Flipping the Conditional: Why We Are Probably Wrong About Probabilities

Abstract

Flipping the conditional is an error of reasoning that occurs when we inadvertently transpose the terms in a conditional probability. A conditional probability arises when the probability of an event occurring (e.g., a positive test result) depends on another event being true (e.g., the presence of a disease). The ordering of terms in a conditional probability is crucially important, and the consequences of transposing the terms can be severe. The error of reasoning is most easily entertained when events are very rare. Flipping the conditional explains why we sometimes misinterpret clinical and diagnostic tests and also underpins a phenomenon known as the "prosecutor's fallacy". Flipping the conditional occurs when we confuse the sensitivity of a diagnostic test with the positive predictive value. The prosecutor's fallacy involves confusing the probability a person is guilty given the evidence with the probability of the evidence given they are guilty. Perhaps the best-known example of the prosecutor's fallacy is that of UK solicitor Sally Clark, who in 1990 was convicted of murdering her two infant sons. Recently, the prosecutor's fallacy has again been in the news, with respect to the statistical evidence presented at the trial of UK nurse, Lucy Letby. In this article, we define the concept of conditional probability and discuss some examples of flipping the conditional that are relevant to pediatric anesthesia and to medical evidence presented at trial.

Keywords: Bayes' theorem; cognitive bias; probability; prosecutor's fallacy; statistics as a topic.

FIGURE 1

Young America's dilemma: “Shall I be wise and great, or rich and powerful?” Cartoon by Louis Dalrymple, published in Puck magazine June 12, 1901 (J. Ottmann Lith. Co., Puck Building, New York City). Image in the public domain. File held the Library of Congress Catalog: https://lccn.loc.gov/2010651418.