The Monty Hall Problem

Welcome to the Show!

The bright lights dazzle your eyes as the stage opens and the cameras roll. A host greets you with a seemingly fluorescent smile  and introduces you to the crowd, his every word dripping with charisma.

“Welcome ladies and gentlemen! I’m Monty Hall, and today we’re offering some BIG prizes!”

A curtain opens behind him and three seemingly innocuous doors are revealed. Monty starts to explain the rules:

“Now dear viewers, here we have three doors. Behind one of these doors is a BRAND NEW TESLA MODEL S!”
*pause for cheering*
“Unfortunately, behind the other two doors there is only a goat. You won’t get many ‘baahs’ to the gallon with one of those!”
*the audience groans*
“But anyway folks, all our guest has to do to win a smashing prize is to pick one of the doors!”

You feel your forehead bead with sweat as the spotlight swings to face you. You’ve always dreamt of owning a shiny Tesla. You strain your ears to try to hear any bleating behind the doors, but alas, the murmur of the crowd is too intense. Prompted by the earnest expression of the host, you raise a shaky finger and point at the centre door.

“The middle door! An interesting choice my friend!”
Monty strokes his chin with feigned concern.
“An interesting choice indeed. I’ll tell you what, pal. I’m going to try and make things a bit easier. I’m going to open one of the doors with a goat behind them. Then, you’ll tell me whether you want to stick with your first choice, or switch to the other door. Sound good?”

Before you get a chance to answer, Monty leaps over to the right-most door and opens it, revealing a black-and-white patched goat who is happily munching on some grass. The audience “aww”s. Monty returns to your side.

“Here’s the million dollar question, buddy. Do you switch, or do you stay?”

 

The Problem with the Problem

This deceivingly simple problem has plagued mathematical thinkers since it was posed in 1975. The Monty Hall Problem is a relatively simple probability exercise: when you pick the first door, you have a 1 in 3 chance of choosing the door with the car behind it. When only two doors remain, intuition would say there is a 1 in 2 chance that the door you picked has a car behind it, because it must be one of those two doors, so it shouldn’t matter whether you switch or not. But here are the facts: if you switch, you have a 2/3 chance of winning the car. If you stay, you only have a 1/3 chance of winning. So the better option is always to switch.

The reason it works this way is because after the first door has been chosen, and there is a 1/3 chance of picking the car, there is no reason that the probability should update when another door is opened. Keep in mind that the host knows where the car is, and will always open a door with a goat behind it. Since there was a 2/3 chance that there is a car behind either of the remaining doors, when one is revealed to be a goat we do not learn anything new about the position of the car, so there is a 2/3 chance that the car is behind the door that you did not choose.

If you’re struggling to get your head around it, you’re not alone. After the solution was published in Marylin vos Savant’s “Ask Marilyn” column in Parade magazine in 1990, around 10,000 readers, including 1000 with PhDs, wrote into the magazine to tell Marylin that she was wrong.  Even prolific mathematician Paul Erdős remained unconvinced until he was shown a computer simulation.

The problem behind the Monty Hall Problem is that our own intuition gets in our way. In one study where this scenario was tested, only 13% of ‘contestants’ chose to switch doors. We can try to circumvent this psychological hang-up by considering the case that there are 100 doors, only one of which hides a car. When you guess initially, there is only a 1 in 100 chance that you were correct in your choice. The host then opens 98 of the doors, knowing that they are hiding goats. Do you really think you were that lucky to have chosen the one door with the car? Or do you switch doors, knowing that if you had chosen a dud then the car has to be behind the unopened door.

If you’re still feeling puzzled, you can give it a try with this Monty Hall Simulator. Will you win a car, or take home a fluffy goat? Either way sounds like a win to me!

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