Twenty-one years ago, a reader of Parade magazine sent in a seemingly simple probability question to Marilyn Vos Savant’s “Ask Marilyn” column. The problem was as follows:
Suppose you’re watching a game show whereby the contestant is asked to choose one of three doors. Behind one of those doors is a prize (a car) while behind either of the other two doors is a goat. The contestant picks one of the doors, but before opening it, the game show host tells the contestant to stop, then opens one of the remaining two doors, revealing a goat. The host then offers the contestant the opportunity to switch doors.
If your answer is that it doesn’t matter either way, since after an incorrect door is opened, two unopened doors remain and therefore the contestant’s chances are fifty-fifty of winning the prize, you’re in good company. Like nearly everyone who has encountered this problem for the first time, you are, however, wrong. If the contestant wants to maximize their chances of winning the prize, they should always switch away from their initial choice and pick the remaining unopened door. If they do, they have a two-thirds chance of winning the car. If they stick with their initial choice, their chances are only one in three, the same as it was before the host revealed an incorrect door. Confused yet?
It may help to consider that, initially, it is clear that the contestant had a one-third chance of choosing the door with the car behind it. Having made that choice, the fact the host reveals additional information shouldn’t affect the probability of the contestant winning the car. This may help clarify why it is good for the contestant to switch doors. But then why does the contestant’s chances of winning jump to two in three rather than fifty-fifty from a switch? The answer lies in the fact the host reveals an incorrect door that the contestant did not initially choose.
Let’s break it down. The contestant’s initial choice could either have been the car, goat #1 or goat #2. If it was either of the goats, of which there is a two-thirds chance, then the host would have revealed the other goat and the contestant wins the car from switching. If the contestant’s initial choice is the car, of which there is a one-third chance, then the host would have revealed either goat #1 or goat #2 and the contestant doesn’t win the car from switching. Two-thirds chance of winning for switching, one-third for staying put.
Still confused? You’re still in good company. In fact, better company than you may realize.
After the answer was published in Parade, no less than ten thousand readers wrote letters protesting that Vos Savant’s answer was clearly wrong. Of those, nearly one thousand had PhDs. Yep, you read that correctly. Not dozens, not hundreds: almost a thousand.
Which brings me to the central point of this anecdote: sometimes common sense fails people spectacularly. Moreover, sometimes common sense spectacularly fails people with PhDs. Why is this so?
Well, for one thing, in the case of the people with advanced degrees (some of which were in Math) we can partially chalk it up to academic specialization. Since these academics weren’t trained in probability specifically, they weren’t used to the highly unintuitive thinking that goes along with understanding the Monty Hall problem. Of course, this didn’t seem to stop them from being certain of their incorrect answers, since, of the letter writers, PhDs were overrepresented ten to one relative to the general population. With articles like “The Way They Were: Celebrity Breakups” it’s unlikely to be a mere by-product of Parade’s PhD-laden readership...
But probably the most important reason for this grand failure in thinking is that our brains haven’t evolved to be good at understanding probability. In our ancestral hunter-gatherer communities, it’s unlikely that any situation would have arisen whereby a deep understanding of probability would have been important for survival. What evolved instead was our reliance on heuristics, or rules of thumb, in decision-making. These were intellectual short-cuts that saved time and energy and were accurate almost all of the time anyway. Unfortunately, in today’s modern mega-societies our reliance on heuristics leaves us prone to thinking about things like the Monty Hall problem incorrectly. So how can we do better?
Well, if there’s one area of study that teaches us to think more correctly, it would be critical thinking. The word is thrown around a lot in the introduction to various University classes but the only one likely to advance one’s ability in that area is its namesake course (PHIL 120). Aside from that, a little self-study can go a long way. Take some time to learn about logical fallacies, cognitive biases, and problems like the Monty Hall problem, and you’ll find yourself fooled less badly and less often.
