The Monty Hall problem is a counter-intuitive statistics puzzle:
- There are 3 doors, behind which are two goats and a car.
- You pick a door (call it door A). You’re hoping for the car of course.
- Monty Hall, the game show host, examines the other doors (B & C) and opens one with a goat. (If both doors have goats, he picks randomly.)
Here’s the game: Do you stick with door A (original guess) or switch to the unopened door? Does it matter?
Surprisingly, the odds aren’t 50-50. If you switch doors you’ll win 2/3 of the time!
Today let’s get an intuition for why a simple game could be so baffling. The game is really about re-evaluating your decisions as new information emerges.
Play the game
You’re probably muttering that two doors mean it’s a 50-50 chance. Ok bub, let’s play the game:
Try playing the game 50 times, using a “pick and hold” strategy. Just pick door 1 (or 2, or 3) and keep clicking. Click click click. Look at your percent win rate. You’ll see it settle around 1/3.
Now reset and play it 20 times, using a “pick and switch” approach. Pick a door, Monty reveals a goat (grey door), and you switch to the other. Look at your win rate. Is it above 50% Is it closer to 60%? To 66%?
There’s a chance the stay-and-hold strategy does decent on a small number of trials (under 20 or so). If you had a coin, how many flips would you need to convince yourself it was fair? You might get 2 heads in a row and think it was rigged. Just play the game a few dozen times to even it out and reduce the noise.
Understanding Why Switching Works
That’s the hard (but convincing) way of realizing switching works. Here’s an easier way:
If I pick a door and hold, I have a 1/3 chance of winning.
My first guess is 1 in 3 — there are 3 random options, right?
If I rigidly stick with my first choice no matter what, I can’t improve my chances. Monty could add 50 doors, blow the other ones up, do a voodoo rain dance — it doesn’t matter. The best I can do with my original choice is 1 in 3. The other door must have the rest of the chances, or 2/3.
The explanation may make sense, but doesn’t explain why the odds “get better” on the other side. (Several readers have left their own explanations in the comments — try them out if the 1/3 stay vs 2/3 switch doesn’t click).
Understanding The Game Filter
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.
But… but… shouldn’t two choices mean a 50-50 chance?
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.
The more you know…
Here’s the general idea: The more you know, the better your decision.
With the Japanese baseball players, you know more than your friend and have better chances. Yes, yes, there’s a chance the new rookie is the best player in the league, but we’re talking probabilities here. The more you test the old standard, the less likely the new choice beats it.
This is what happens with the 100 door game. Your first pick is a random door (1/100) and your other choice is the champion that beat out 99 other doors (aka the MVP of the league). The odds are the champ is better than the new door, too.
Visualizing the probability cloud
Here’s how I visualize the filtering process. At the start, every door has an equal chance — I imagine a pale green cloud, evenly distributed among all the doors.
As Monty starts removing the bad candidates (in the 99 you didn’t pick), he “pushes” the cloud away from the bad doors to the good ones on that side. On and on it goes — and the remaining doors get a brighter green cloud.
After all the filtering, there’s your original door (still with a pale green cloud) and the “Champ Door” glowing nuclear green, containing the probabilities of the 98 doors.
Here’s the key: Monty does not try to improve your door!
He is purposefully not examining your door and trying to get rid of the goats there. No, he is only “pulling the weeds” out of the neighbor’s lawn, not yours.
Generalizing the game
The general principle is to re-evaluate probabilities as new information is added. For example:
A Bayesian Filter improves as it gets more information about whether messages are spam or not. You don’t want to stay static with your initial training set of data.
Evaluating theories. Without any evidence, two theories are equally likely. As you gather additional evidence (and run more trials) you can increase your confidence interval that theory A or B is correct. One aspect of statistics is determining “how much” information is needed to have confiidence in a theory.
These are general cases, but the message is clear: more information means you re-evaluate your choices. The fatal flaw of the Monty Hall paradox is not taking Monty’s filtering into account, thinking the chances are the same before and after he filters the other doors.
Here’s the key points to understanding the Monty Hall puzzle:
- Two choices are 50-50 when you know nothing about them
- Monty helps us by “filtering” the bad choices on the other side. It’s a choice of a random guess and the “Champ door” that’s the best on the other side.
- In general, more information means you re-evaluate your choices.
The fatal flaw in the Monty Hall paradox is not taking Monty’s filtering into account, thinking the chances are the same before and after. But the goal isn’t to understand this puzzle — it’s to realize how subsequent actions & information challenge previous decisions. Happy math.
Let’s think about other scenarios to cement our understanding:
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.
Monty goes wild
Monty reveals the goat, and then has a seizure. He closes the door and mixes all the prizes, including your door. Does switching help?
No. Monty started to filter but never completed it — you have 3 random choices, just like in the beginning.
Monty gives you 6 doors: you pick 1, and he divides the 5 others into a group of 2 and 3. He then removes goats until each group has 1 door remaining. What do you switch to?
The group that originally had 3. It has 3 doors “collapsed” into 1, for 3/6 = 50% chance. Your original guess has 1/6 (16%), and the group that had 2 has a 2/6 = 33% of being right.