Understanding the Monty Hall Problem

Suppose you're on a game show and you're given the choice of three doors. Behind one door is a car; behind the others, goats. The car and the goats were placed randomly behind the doors before the show. The rules of the game show are as follows: After you have chosen a door, the door remains closed for the time being. The game show host, Monty Hall, who knows what is behind the doors, now has to open one of the two remaining doors, and the door he opens must have a goat behind it. If both remaining doors have goats behind them, he chooses one randomly. After Monty Hall opens a door with a goat, he will ask you to decide whether you want to stay with your first choice or to switch to the last remaining door. Imagine that you chose Door 1 and the host opens Door 3, which has a goat. He then asks you "Do you want to switch to Door Number 2?" Is it to your advantage to change your choice?

The answer is, it is always better to switch doors.

Door 1	Door 2	Door 3	Chosen	Shift	No Shift
Goat	Goat	Car	1	Car	Goat
Goat	Goat	Car	2	Car	Goat
Goat	Goat	Car	3	Goat	Car
Goat	Car	Goat	1	Car	Goat
Goat	Car	Goat	2	Goat	Car
Goat	Car	Goat	3	Car	Goat
Car	Goat	Goat	1	Goat	Car
Car	Goat	Goat	2	Car	Goat
Car	Goat	Goat	3	Car	Goat

The above table gives all the possible combinations. From the table, it is clear that, if we switch the door, we get car in 6 out of 9 combinations. If we don't switch, we get car only in 3 out of 9 combinations. Detailed description can be found at Wikipedia.

The explanation on BetterExplained is much better. It explains the logic behind that in words.

Let’s see why removing doors makes switching attractive. Instead of the regular game, imagine this variant:

* There are 100 doors to pick from in the beginning
* You pick one door
* Monty looks at the 99 others, finds the goats, and opens all but 1

Do you stick with your original door (1/100), or the other door, which was filtered from 99? (Try this in the simulator game; use 10 doors instead of 100).

It’s a bit clearer: Monty is taking a set of 99 choices and improving them by removing 98 goats. When he’s done, he has the top door out of 99 for you to pick.

Your decision: Do you want a random door out of 100 (initial guess) or the best door out of 99? Said another way, do you want 1 random chance or the best of 99 random chances?

We’re starting to see why Monty’s actions help us. He’s letting us choose between a generic, random choice and a curated, filtered choice. Filtered is better.

Overcoming Our Misconceptions

Assuming that “two choices means 50-50 chances” is our biggest hurdle.

Yes, two choices are equally likely when you know nothing about either choice. If I picked two random Japanese pitchers and asked “Who is ranked higher?” you’d have no guess. You pick the name that sounds cooler, and 50-50 is the best you can do. You know nothing about the situation.

Now, let’s say Pitcher A is a rookie, never been tested, and Pitcher B won the “Most Valuable Player” award the last 10 years in a row. Would this change your guess? Sure thing: you’ll pick Pitcher B (with near-certainty). Your uninformed friend would still call it a 50-50 situation.

Information matters.

Your buddy makes a guess

Suppose your friend walks into the game after you’ve picked a door and Monty has revealed a goat. But he doesn’t know the reasoning that Monty used.

He sees two doors and is told to pick one: he has a 50-50 chance! He doesn’t know why one door or the other should be better (but you do). The main confusion is that we think we’re like our buddy. We forget (or don’t realize) the impact of Monty’s filtering.