A Simple Explanation of the Monty Hall Problem

The Monty Hall problem — based off of the TV Show "Let's Make a Deal" and named after the original host, Monty Hall — is a notorious problem in statistics.

Monty presents to you three closed doors. Behind one is a prize, behind the other two are nothing (I think the original formulation says they're goats — either way, not something you want).

You pick one of the doors.

No matter which you pick, Monty (who knows what's behind every door) opens an empty door, and offers to let you switch your guess to the other, remaining closed door.

Should you switch? Does it matter?

You should indeed switch. Your odds of winning are 2/3 if you do, vs 1/3 if you don't. However, many people find this difficult to grasp at first.

Jeff Kaufman has a post where he talks getting it wrong initially, and how bad humans are at it in general.

"The person who first showed me this problem failed to convince me that I should switch; that took testing it with pennies and cups. ...

"Since then, in various conversations with people who were sure switching didn't help, I've tried many times to describe how the odds for switching can be 2/3. I've come up with many explanations that I think would have convinced me, but they don't seem to convince others."

He's not alone:

"You should switch. However, it is so counterintuitive that I had to write a simulation script with 1M iterations... to confirm indeed switching yields twice the probability of winning than staying." — u/creekwise

"I also wrote a simulation script." — u/Jusque

Here's the clearest way I've found to think about it.

Say you always switch. Now there are two outcomes:

You originally pick one of the two dud doors (2/3 chance). Then Monty will open the other dud (he has to, the only other option is the prize door and he can't open that) and so if you switch to the unopened door, you'll win.

You originally pick the prize (1/3 chance). Monty will open one of the two duds, you'll switch to the other, unopened dud. You won't win.

So: 2/3 of winning if you switch. Don't switch, it all depends on whether you pick the prize initially, which you'll do 1/3 of the time.