Andrew Gilhooley's Blog

I wrote an article about the Monty Hall problem on my old blog several years ago and found myself thinking about it again recently. I was reminded of the importance of the ability to think critically and to be aware of how easily our minds can be fooled into believing something merely because it “feels right”. Nowadays we are bombarded with so much misinformation, disinformation, conspiracy theories and moral panic every day it is easy to lose track of what is real and what is not. The growth of AI has only compounded the problem. What follows is a re-write of my original post.

If you’re not familiar with the Monty Hall problem, it is a logic and probability puzzle named after the host of TV Show “Let’s Make A Deal” and first published in 1975. The puzzle goes something like this: 

Imagine that you have made it through to the final round of a TV game show.  The show’s host shows you three doors. 

One of the three doors, he tells you, hides this evening’s grand prize: a brand-new car (presumably with a bikini-clad girl waving from the driver’s seat window, as always seemed to be the case on these kinds of shows when I watched them as a youngster). Behind the other two doors there is nothing (or in some tellings, a goat). All you have to do to win the car is to pick the correct door.

So you make your choice of door and the host asks you if you are sure of your choice.  You say yes; after all it’s a simple one-in-three chance that you will win, right?

The host, who knows what is behind each door, then walks up to one of the doors that you didn’t choose and opens it, to reveal an empty space.  He then offers you a choice: You can switch your choice of doors between the one you originally chose and the remaining closed door, or you can stick with your original choice.

In the above illustration, you have chosen door number 1. The host has opened door number 3 to show an empty space behind it. Should you change your choice to door number 2 or should you stick?  Which gives you the better chance of winning?  Does it make a difference?

This puzzle has been debated many times over the years, most notably in the US magazine Parade in 1990, when it generated a flood of letters from readers arguing their case as to the correct answer.

Most people choose to stick with their original choice of door, figuring that whether or not you switch makes no difference as there is a 50% chance that you have won the car.  After all, you may have chosen correctly in the first place, so by switching you might be giving away your prize. Sticking with your first choice just feels like the right thing to do. 

Actually, it’s not.  In this game, you should always switch, as you actually will win two times out of three, even though intuitively this might seem wrong. 

To explain why, let’s walk through a round of the game, this time knowing where the car is. Here is the setup at the start of the game. The car is hidden behind door number 2:

Let’s say like in our first example, you choose door number 1. The host opens door number 3 to show that it is empty and gives you the opportunity to switch.

Here, if you choose to switch to door number 2, you will win the car. If you stick with door number 1, you lose.

The same result will occur if you choose door number 3. The host will open door number 1, showing it as empty.

Once again, if you switch your choice to door number 2, you will win, stick with door number 3 and you lose.

Finally, let’s imagine you choose door number 2, which contains the car. The host will open either one of the other two doors as they are both empty.

This time, if you switch your choice you will lose; if you stick with your original choice you will win.

Let’s look at these results in a table:

As you can see, if you don’t switch, your chance of winning is one in three as you might expect. However, switching your choice increases your odds of winning to two times in three, even though it somehow feels like it shouldn’t. Try re-running the game with the car behind a different door; you’ll get the same result.

If you want to think about it another way, consider that at the start of the game you have two chances in three to pick an empty door. The host will always open an empty door, so you have two chances in three to switch to the car.

In his excellent 2006 book Tricks of the Mind, British illusionist Derren Brown spends a chapter discussing the Monty Hall problem at some length, along with several other questions and puzzles that show how easily our thinking can be fooled. If you’re still not convinced by my explanation of the Monty Hall problem above, Brown walks you through another version, this time with a hundred doors.

American comedian Stephen Colbert coined the term “truthiness” to describe the quality of a claim being accepted as true because of gut feeling, or emotion, or a desire for it to be true rather than on facts, evidence or logical examination.

For a recent example of truthiness in operation, US Speaker of the House Mike Johnson had this to say about the June election in California: “They are counting votes weeks after the election.” This is perfectly legal under California law, as all postal votes must be counted so long as they are postmarked by the appropriate date.

He went on to insinuate some kind of malfeasance in the election, saying “I’m not saying it’s rigged. I’m saying it stinks to high heaven, and everybody knows that.”

When pressed for evidence of his claims, he replied “Some of these efforts are so diabolical and so far upstream that it’s impossible to prove, but I think everybody knows instinctively, something is wrong here, and that’s a concern” (emphasis mine). He is effectively saying “I have no proof to back up my claim but I feel like I’m right, so my claim is true.” How many other people heard or read his claim and believed it simply because it felt right to them too?

I remember recently sitting in the waiting room at a full-service car wash where a cable news channel was playing. After just a few minutes I became very aware that they alternated between stories designed to play into the viewers’ sense of outrage and stories designed to play into their sense of fear.

I’m not here to make any kind of political point in this post. My point is simply that we can all easily be fooled by what at first glance seems to make sense. Television, radio, websites and so on can be very persuasive with their claims, whether they are for some wonderful vitamin supplement, a miracle diet or a get-rich-quick opportunity.  News reports will selectively pick statistics and describe them in such a way as to give them sensational meanings. Conspiracy theories, scare stories and moral panics come at us from all angles and spread rapidly via social media. We live in a society that is increasingly disdainful of experts, evidence, science and objective facts. And that’s without throwing AI-generated content into the mix.

To wrap this up, the need to think critically about what we see and hear has never been more important. As someone once said, a lie can be halfway around the world before the truth has a chance to put on its shoes. With the degree of media saturation that we have today, and the ability of that media to manipulate us, we have to remind ourselves that objective truths do exist and that the effort to find those truths is worthwhile.