A Simple Explanation of the Monty Hall Problem

Background

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.

Problem

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?

Answer, which humans are bad at understanding

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

Simple Explanation

Here's the clearest way I've found to think about it. You can go into it with two strategies:

Strategy #1: Don't Switch

Say you go in determined not to switch. No matter what, you're not switching. What happens?
You pick a door. Monty opens another, empty door. This means nothing. No matter which door you picked initially (prize or no prize), there's always at least one other empty door for Monty to show you (he knows ahead of time what's behind everything).
So he shows you an empty door. You don't switch. If you happened to have guessed the right door initially, you get the prize.
There's a 1/3 chance you win with this strategy.
Fine.

Strategy #2: Always Switch

Now say you go into it saying you'll always switch.
What happens? You pick a door, Monty opens another, empty door. Again, no matter what you picked originally (prize or no prize), there's always at least one empty door for Monty to show you (he knows ahead of time what's behind everything).
So shows you an empty door. You switch. What happens?
Well, if you guessed one of the two empty doors originally (it doesn't matter which one), Monty will open the other empty door, and you'll switch to the prize.
Since originally two out of the three doors are empty,
you have a 2/3 of winning if you go into it planning on switching.
The only way you
don't
win is if you pick the prize door originally, then Monty opens an empty door, you switch away to the
other
empty door and walk away with nothing (or a goat). There's a 1/3 this happens.