The Legacy of Monty Hall

I have to confess at the outset that I have never taken any instruction in the subject of mathematical probabilities, as the rest of this piece will no doubt demonstrate.

So it was when I started reading Harvard celebrity intellectual Steven Pinker’s new book Rationality: What it is, Why it seems scarce, Why it matters and saw the Monty Hall problem laid out right on page 19, I knew I had to face up to my failure to understand it if I was going to derive any benefit from the rest of the book.

The Monty Hall problem is taken from the playbook of the once popular television game show Let’s Make a Deal, of which Mr. Hall (he died in 2017, we’ll call him Monty) was the co-creator and emcee. And it has fascinated a generation of mathematics nerds and casual grazers like me. Nowadays, of course, you can go onto YouTube and find all kinds of solutions to the problem, explained with anything from a pen and drawing paper to sophisticated computer simulations. The problem demonstrates how strongly an intuitive answer can take hold and how hard it can be to see the mathematically correct answer.

The show featured a ‘three doors’ game. Behind one of the three doors was a shiny new car. Behind each of the other two was a goat. Contestants had to make a preliminary choice as to which door they wanted to pick, the car being the prize everyone wanted. But the contestant didn’t know which of the three doors contained a goat or a car. Monty, who knew, would next open one of the two doors not selected by the contestant, provided it harboured a goat, leaving the car prize to be discovered behind one of the two unopened doors. (Monty would have had one or two doors to choose from, depending upon whether the contestant had unknowingly selected a goat or the car as their preliminary choice). The contestant would then be put to a choice before a second and final door was opened. Did they wish to stick with their original choice of door, or did they wish to change their door to the one they had not chosen initially and that had not been chosen by Monty. They were now armed with the information about the precise location of one of the two goats. Should that make any difference?

The intuitive answer is that it shouldn’t. Whichever door is selected, it remains the case that behind one door is a car and behind the other a goat. It’s a 50-50 chance which one you pick. Why change for the sake of change?

But it’s not that simple. In fact, you are better off to switch doors. By sticking with your original door choice, you would be wasting the knowledge that was given to you by Monty’s revelation of the goat door. You would be settling for a one in three chance that you had picked the door behind which was a car. Switching doors, however, ups your likelihood of success to two in three.

It almost defies comprehension, but here’s how it goes. Let’s say you had door number one selected as the preliminary choice for the location of the car. If door number three was opened by Monty, you would move your guess as to the car’s location to door number two. You would lose if the car were in fact located behind door number one. However, you would win if it were located behind door number two. You would also win if the car were located behind door number three, because Monty would have had to reveal the goat in gate number two. (Gate number one was the one you were sitting on from your preliminary choice, while gate number three contained the car). So in two out of three scenarios, you come out the winner with the switching strategy.

Note that this manoeuvre doesn’t guarantee the result of any particular game, just as flipping a coin 10 times and coming up heads 10 times in a row doesn’t have any value in predicting what the outcome of the next flip will be. What it does tell you is that over, say, a hundred games you would show a track record of about two-thirds of your games justifying your move to the new door. Useful to know, especially if you don’t want to get into the goat herding business. And a contestant on Let’s Make A Deal only gets to play the game once, so it presents to them as a tossup: let’s cut them a little slack.

Even after demonstrating to myself that the switch option is the better one, there are moments when I revert to the tossup scenario and lose track of the switch option’s logic—such is the power of the mind to go for the intuitive.

Thank you, Monty, for leaving us such a delightfully challenging legacy. You will be pleased to know that I have resolved to take Mr. Pinker’s advice and try to make more rational decisions in my own life. What’s the probability of that happening? I hope the answer is not counter-intuitive.