The Monty Hall Problem

A great many of you remember the game show, “Let’s Make a Deal,” and many of you are probably familiar with “The Monty Hall Problem,” named for the show’s host.

This is a logic problem that seems to have an illogical answer, but I finally figured out a way to understand it.

Let’s say you are a contestant on “Let’s Make a Deal.” Monty tells you that behind one door is a new car, but behind each of the other two doors is a goat. You have to choose one door. We can all agree that your odds are 1 in 3 of choosing the door with the new car. For argument’s sake, let’s say you choose door #3.

Whether or not you chose the door which hides the car, there is a 100% chance that there is a goat behind at least one of the other two doors. Monty then opens one of those doors, revealing a goat. In this case we’ll say it’s door #1. Monty then offers you the opportunity to change your door from door #3 to door #2. Should you do so?

The answer is “Yes.” Your odds will increase from 33% to 67% if you switch.

When I first heard about this, I could not wrap my brain around it. In my mind, there is a 1 in 3 chance I chose the correct door, and since Monty always had a door with a goat behind it to challenge my initial choice, I couldn’t see how my odds would change once Monty revealed a goat. So I actually played it out, and that’s when it all made sense.

If I initially chose the correct door - a 33% chance - I would always lose once I changed doors after being shown the goat.

But if I initially chose one of the two wrong doors - a 67% chance - I would always switch to the correct door after being shown the goat.

Therefore, my odds switch from 33% to 67% of correctly choosing the new car.

So back to our example, if you chose door #3 and the car was actually behind door #3, you would lose after switching. But if you chose door #3 and the car was behind either door #1 or door #2, you would be guaranteed to choose the winning door after the goat was revealed and you switched doors.

Here’s a little of Monty’s dealmaking. Also, open thread.

[buck.throckmorton at protonmail dot com]

posted by Buck Throckmorton at 08:06 PM

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