How To Win A Car With Monty Hall Math

Many of us grew up watching Monty Hall on Let’s Make a Deal. It turns out that the show presents something of a mathematical conundrum: does switching your choice at the last minute between the two remaining rooms increase your chances of success? The video below presents the answer and it may surprise you.

The so-called Monty Hall paradox is a veridical paradox that is similar to the Three Prisoners problem and a less known Bertrand’s box paradox.

The paradox was first presents by Steve Selvin in a publication in the American Statistician in 1975. He argued that a player who stays with the initial choice wins in only one out of three times while a player who switches wins in two out of three times. Here is another explanation from a math site:

Imagine that Monty opens a door and shows that there’s only a goat behind it. Consider that the car is more likely to be behind a door other than the one you choose. Monty has just shown that one of those two doors – which together have the greater probability of concealing the car – actually conceals a goat. This means that you should definitely switch doors, because the remaining door now has a 2/3 chance of concealing the car. Why? Well, your first choice still has a 1/3 probability of being the correct door, so the additional 2/3 probability must be somewhere else. Since you know that one of the two doors that previously shared the 2/3 probability does not hide the car, you should switch to the other door, which still has a 2/3 chance of concealing the car.

The idea is so counterintuitive that hundreds of experts wrote into the magazine contesting the theory. However, this video from 2009 shows the theory in execution.

I will be trying this tonight with the kids as part of the Turley Blog Research Center.

24 thoughts on “How To Win A Car With Monty Hall Math”

  1. I would note that they tested this in a recent Mythbusters episode. Adam and Jamie did 100 trials with Adam always switching and Jamie always sticking with his original choice which was pretty convincing evidence that the 1/3, 2/3s math is correct, but the really interesting thing that they found was that in 20 trials with ordinary people ALL of them stayed with their original choice–proving beyond a doubt that if Monty has complete information and a keen grasp of human psychology (and especially if he is allowed to NOT give a choice), then the show isn’t going to pay out much more than he wants it to (there must be winners or else no one will watch…).


    The Mythbusters also addressed yawning as a contagion, but they didn’t address a virtual vector… *yawn* 😉

  2. Leonard Mlodinow talked about this as part of a discussion of the law of sample space in his book, “The Drunkard’s Walk: How Randomness Rules Our Lives”. The whole book is a great read.

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