The Curious Incident from Let’s Make a Deal
Statistical analyses depend on probability distributions. So it is not surprising that rudiments of probability theory are emphasized in statistics courses. The theme of this post is that, for many, probability theory is not being taught well, not being understood, or both.
Some years ago, I read a book called The Curious Incident of the Dog in the Nighttime, by Mark Haddon. Haddon, and his protagonist in the book, are fascinated with mathematics. One of the reasons given for this fascination is that math can explain things that otherwise might seem counterintuitive. An example of this power of math, given in the book, is the “Let’s Make a Deal” scenario. Perhaps you know of this scenario and it implications, but reading this book was my first exposure to it. It goes like this:
The host of Let’s Make a Deal has great news for you. You could win money! All you must do is choose one of the three doors: A, B, or C. Behind one of those doors is money. Behind the other two doors, goats. Now goats are cool and all, but let’s assume you want the money. So you choose a door, say A. Before the host opens door A, he first opens one of the other doors. The host, of course, knows which door hides the money, so will only open a door hiding goats. So the host opens a door (B or C) to reveal goats. Now, the host says you can keep door A, or switch to the other unopened door. What do you do? One door must hide the money. The other must hide goats. 50/50, right? So it doesn’t matter if you switch or keep door A, right?.
Probability theory, however, shows that given this scenario, switching increases the chances of getting the money beyond 50%. To be exact, switching gets the money 2 in 3 times, or about 66.7% of the time. Not switching, then, gets the money 1/3 times. When the host asked you to pick among 3 doors, that choice created a 1/3 chance of picking the money and 2/3 chance of picking goats. The act of opening another door that hides goats does not change your winning probability (and surely does not raise it to 50%). However, 2/3 of the time (i.e., when you first chose a door that hides goats), there is only one door the host can open that hides goats. Thus, 2/3 of the time, the remaining door (i.e., not the one you originally chose, nor the door the host just opened) hides the money.
This Let’s Make a Deal exercise in probability is not very complicated, but I have been amazed by the number of people, many with advanced degrees, who have argued with me that there is no benefit to switching. My explanation usually falls on deaf ears, suggesting I did not explain well, or the message just is not being received. At that point, I turn to Google, or this page: https://en.wikipedia.org/wiki/Monty_Hall_problem.