What Everybody Gets (Slightly) Wrong About the Monty Hall Problem
Admittedly, when I first heard the Monty Hall problem, I didn’t believe it.
Why would switching your decision help your odds if your choice is random? Should I be constantly switching lanes at the grocery store? What did this mean for game shows like Deal or No Deal with dozens of doors?
At the time, I wasn’t able to explain it well. Even as others explained the Bayesian logic behind it, something didn’t seem right.
When I tried to replicate it with programming code, as others said should prove the theory, I wasn’t able to.
Eventually I realized it: The Monty Hall problem assumes that when a door is revealed the first time, it is a loser every time. It’s always farm animals rather than a sports car or a bag of money.
That might be how the brain teaser is set up, but that’s not how the game show worked. In Let’s Make a Deal, the first door revealed could have been a sports car or a bag of money or a trip to the Bahamas. Otherwise, there was no tension.
If every time they said, “we’re going to show you what was behind Door #1, which you didn’t choose, and it’s going to be a donkey,” it would be a waste of the audience’s time. There’s no anticipation that behind Door #1 could be a bag of money. They would just open the door quickly, show the donkey, and move on to the actual reveal of importance.
There was always that anticipation that *hopefully* there would be a donkey behind Door #1 so the bag of money was still in play.
So maybe that’s why the Monty Hall problem is called the Monty Hall problem. It’s not the Let’s Make a Deal problem.
Knowing all of that, the Monty Hall problem becomes easier to understand.
One way I think of it is that if you always chose Door #1 and never switched, it’s a 1/3 probability. But if you do switch after a door is revealed to be a loser, then you are simply choosing between two options where it’s a 1/2 probability.
In the situation where the first door revealed could be a bag of money, you don’t get that chance to switch. So even if you were planning on switching, you don’t get the chance because you already lost.
And that may not be a great explanation. Which is why it’s still a good brain teaser, but it’s not how the show worked.