Statistically Speaking

Episode 1: And behind Door #1… This slideshow is licensed under a Creative Commons Attribution-NoDerivs 3.0 Unported License. Please contact the author for additional permissions.

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Let’s say that you’re on a game show, and the host takes you to a wall

with three closed doors.

Behind one of those doors, he says, is…

1 2 3

AN AWESOME RED SPORTS CAR NAMED AFTER A VENOMOUS SNAKE!

Car image by BrokenSphere on Wikimedia Commons. Snake image by Kamalnv on Wikimedia Commons. Used under Creative Commons license.

But there are two other doors, and behind each of those is a less enticing prize…

A YEAR’S SUPPLY OF VENOMOUS SNAKES, DELIVERED STRAIGHT TO

YOUR BED!Viper image by Harold van der Ploeg on Wikimedia Commons. Used under Creative Commons license.

All you’ve got to do is make the right choice.

So, will you pick Door #1, Door #2, or Door #3?

1 2 3

It might help to first understand your chances of winding up with a cool car

instead of a bed full of snakes.

With all three doors unopened, your chances are 1 in 3 that you’ll drive away in

the car today, and 2 in 3 that you’ll be crawling out of bed with a snake bite

tomorrow.

In terms of the probabilities of choosing incorrectly, your best bet is to refuse to open

any of the doors.

But, you’re a crazy gambler with a love of narrowly avoiding certain death, so you hold

your breath and choose door #1.

And then the host says something surprising…

1 2 3

“I’m going to show you what’s behind one of the doors you didn’t pick,” he says.

And as door #3 opens, a bunch of snakes pour out onto the stage!

1 2 3

“Now that you know what’s behind that door,” he says, “do you want to change your

choice to door #2?”

And if you’re smart, you will.

1 2 3

?

“Huh?” you might be saying. “Aren’t my odds 50/50 now?”

I used to think so, too… until I examined the probabilities a little more closely.

1 2 3?

50% 50% 0%

In the first step of the game, your

chances were one in three of making the right choice.

In the second step, your chances are still one in

three.

The only difference is that we know what’s

behind door #3.

1 2 333% 33% 33%

1 2 3?

33% 33% 33%

You see, the host didn’t choose a door at random. He knows what’s behind all three

doors, and he’s playing up the drama.

And this means that the second choice you have to make is dependent on the first one.

1 2 3?

It might seem crazy, but knowing what was behind door #3 after you’ve chosen doesn’t

even the odds.

Remember - you initially only had a one in three chance of choosing the right door.

In other words, the odds were initially in favor of you making the WRONG choice.

Therefore, with one bed of snakes out of the way, it’s more likely that the second door will

be the one with the cool car.

So, when you get to the second step, and the host asks you if you’d like to change,

the odds favor making the switch.

1 2 3

It sounds crazy, I know, but if you try it out with a friend or with a computer model,

using the same rules, you’ll find that you win about twice as often if you switch.

(Don’t be alarmed if you’ve found this exercise surprising… even mathematicians have famously had trouble with this one!*)

*I’m not kidding, either. Marilyn vos Savant has had plenty of issues with this one after printing the correct answer in her column.

To summarize:

Your chances of having selected the right door initially are 1 in 3.

Your chances of selecting the right door if you switch are 2 in 3, because one dud is

eliminated, and you were more likely to guess wrong initially.

By switching doors at the second step, you are twice as likely to win!

Would you like to learn more?

I recommend the book The

Drunkard’s Walk: How Randomness Rules Our Lives

by Leonard Mlodinow.*

*Check out Chapter 3 for a thorough discussion of this topic.