Right for the wrong reasons: common bad arguments for the correct answer to the Monty Hall Problem

The answer to the Monty Hall Problem that many people—including some well-trained mathematicians—initially give is incorrect. Nonetheless, there is little controversy among mathematicians and philosophers about what the correct answer is. However, many different arguments have been given for this answer. Although Bayes’s Theorem is the gold standard for carrying out probabilistic inferences, many mathematicians and philosophers try to give shorter and more intuitive arguments for the correct answer to the Monty Hall Problem. Unfortunately, as we argue in this paper, an unconscionably large number of these shortcut arguments involve bad reasoning. Thus, many people end up believing “the right answer for a wrong reason.” Moreover, since these arguments only yield the correct answer in a very restricted range of cases, people learn techniques that lead to false conclusions when they are applied to many other probabilistic inference problems. In this paper, we identify three distinct bad arguments for switching in the Monty Hall Problem that are commonly given by quite reputable sources. We show that these arguments yield incorrect answers when applied to slight variations of the Monty Hall Problem. And we identify exactly where each argument goes wrong. We argue that it would be much better to simply teach people to ask how likely the evidence is given each of the hypotheses.