The Monty Hall problem and other paradox puzzles

May 29, 2017 5:19 pm Last Updated: May 29, 2017 5:19 pm

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, 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?

It’s a problem that’s stumped people consistently for decades, because even when you learn the answer it sounds like it couldn’t possibly be true.

It’s called the Monty Hall problem, named for the game show host Monty Hall (of “Let’s Make a Deal”). The puzzle was original posed by American statistician Steve Selvin in 1975 through a magazine.

His answer was that yes, you should switch your answer, and by doing so will have a better chance of winning the car.

At the beginning of the game, the player does not know what is behind any of the doors, giving any door a 1/3 chance of holding a car.

Once Door 3 is opened, most people assume that Door 2 and Door 1 each have a 50 percent chance of hiding a car and therefore there is no advantage to switching.

Most of us, including the magazine readers the first time around, would be wrong.

It’s standard probability, and one simple explanation is this:

If Door 1 (the one you initially picked) has a 1/3 chance for the car, then Doors 2 and 3 together have a 2/3 chance, and once Door 3 has been eliminated, there is a 2/3 chance that Door 2 hides the car, and still a 1/3 chance that Door 1 hides the car.

If this makes you want to bang your head on your desk, you are not alone: about 10,000 readers (including about 1,000 PhD holders) of the original magazine where this question was posed wrote in to say that the answer was wrong.

But if you like paradoxical seeming puzzles, check out the Three Prisoners problem and the Bertrand’s box paradox.