An Understandable Solution To The Monty Hall Problem

Charlottesville

August 1, 2024

If you are not familiar with the Monty Hall Problem, here’s a statement of it:

Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say Door A and the host, who knows what's behind the doors, opens another door, say Door C, which has a goat. He then says to you, "Do you want to pick door Door B. Is it to your advantage to switch your choice?

If you are like me, your intuition is that your odds of picking the correct door of the three is 1/3. After the host reveals what’s behind one of the two doors you did not pick, your odds of having picked the correct door are now 1/2 — meaning there is no advantage to switching — so you don’t.

And you would be wrong. By switching you increase your odds of winning from 1/3 (not 1/2) to 2/3.

Don’t believe me? Work yourself through the decision tree above.

The Internet of full of explanations of the solution to this problem. Most of them rest on Laplace’s formula of Bayes’ Theorem. (In words, Bayes’ Theorem states “The conditional probability of an event A, given the occurrence of another event B, is equal to the product of the event of B, given A and the probability of A divided by the probability of event B.”) I find those explanations difficult to follow — which probably explains in part why I have trouble with probabilities! But decision trees are in my wheelhouse, and I found using one here really helped me understand the solution (and find the correct answer).

I’m fascinated by how for many of us, our intuitions on what appears to be a straight-forward solution can be wrong. Kahnamen points out that our minds just aren’t good at quickly processing problems that involve probability. Our intuition kicks in, we quickly come up with a solution that makes sense, and we stop thinking satisfied that we must be right. Our minds tell us we don’t need to do the hard work of understanding Bayes’ Theorem or developing a decision tree because it’s obvious we are correct — our minds, generally speaking, don’t like working hard especially when our intuition is so sure of itself.

Hmmm. Don’t believe everything you think?? Seems I have seen that someplace before. 🤪