The Monty Hall problem (also called the Monty Hall paradox) is a famous probability puzzle and brain teaser loosely based on the american television show Let’s Make A Deal.

If you haven’t heard of this puzzle before you’ll probably have heard a different, mathematically equivalent variation to it.The **‘Three prisoners problem’ **is another popular probability puzzle which bears a lot of similarity to the Monty Hall problem.

The Monty Hall problem is a brain teaser which first received popularity when a published statement of the problem was published in Marilyn Vos Savant’s “Ask Marilyn” column in Parade magazine in 1990. Parade magazine claimed that Marilyn Vos Savant had the highest IQ in the world according to the Guinness book of world records hall of fame. The column usually featured her answering complicated maths questions sent in by readers.

In September 1990 a man named Craig F. Whitaker of Columbia, Maryland posed this question:

“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 No. 1 [but the door is not opened], and the host, who knows what’s behind the doors, opens another door, say No. 3, which has a goat. He then says to you, “Do you want to pick door No. 2?” Is it to your advantage to switch your choice?”

Vos Savant’s response was that the contestant should always switch to the other door. This is because the chances are 2 in 3 that there will be a car behind that door.

Hang on a minute. This sounds erroneous. If you use your intuition you would immediately think that the chances are 50:50 because there is an equal chance that the car would end up behind any door. This is because the host has evicted one of the doors, leaving two doors left for you to decide. Hence, the probability that the car should be behind any one of those two doors should be 50:50.

Therefore, many readers refused to believe that switching was beneficial as loads of people stuck with their intuition and postulated that the chance was equal. They were eager to ridicule Marilyn Vos Savant, the so-called ‘Highest IQ in the world’ holder.

After the Monty Hall problem appeared in *Parade*, approximately 10,000 readers, including nearly 1,000 with PhDs in mathematics, wrote to the magazine claiming that Vos Savant was wrong. Many of them reacted in disgust, one of them stated that:

There is enough mathematical illiteracy in this country, and we don’t need the world’s highest IQ propagating more. Shame! – Scott Smith PH.D University of Florida.

However, it turns out that Marilyn Vos Savant was right. And there are many different ways you can show this. You can present the answer mathematically but I think it would be better if I presented it in diagrammatically.

This rather badly self-made Monty Hall tree diagram should give you the basic idea. Whatever decision you make, 2 times out of 3 you get a car. If you stick with your decision, (so you **don’t change**). You will only get a car 1 time out of 3.

The reason I wanted to share this with you guys was that I like the story behind this problem. It shows that sometimes your intuition can be wrong and that there are different ways of exploring puzzling conundrums. It also shows that people who have PhDs aren’t always right. Which is nice to know.

Great post! A perfect example of the failures of human intuition when applied to an unusual problem. Bayesian probability triumphs!

We’re thinking “random choice, random choice, random choice,” and forgetting that Monty’s choice isn’t random. He’s always going to show you a goat. If he was making a random choice he’d show you the car 1/3 of the time, and the game would be over.

Exactly Dave, It’s a classic example that sometimes, logic just triumphs. And TheStevenator: Yes, Bayesian probability never fails to disappoint does it?

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