Please Explain (actually explain) The Monty Hall Problem

[PeroK]

I thought of another angle. Imagine the game is played simultaneously by three players, each with their own instance of the game. In each game the car is behind the same random door for all three players.

The first player always chooses door 1 and sticks; the second player always chooses door 2 and sticks; and, the third player always chooses door 3 and sticks.

If stick wins 50% of the time, then each player must win 50% of the time, and the car must be behind each door 50% of the time. Which is impossible.

 

[Ibix]

If stick wins 50% of the time, then each player must win 50% of the time, and the car must be behind each door 50% of the time. Which is impossible.

I really like this one. I realise it's the "stick only wins if you guessed right first time" explanation turned round a bit, but it explicitly forces you to fit three things evenly into two boxes if you want to believe 50/50. Neat.

 

[FactChecker]

I thought of another angle. Imagine the game is played simultaneously by three players, each with their own instance of the game. In each game the car is behind the same random door for all three players.

The first player always chooses door 1 and sticks; the second player always chooses door 2 and sticks; and, the third player always chooses door 3 and sticks.

If stick wins 50% of the time, then each player must win 50% of the time, and the car must be behind each door 50% of the time. Which is impossible.

But there might be a detail missing here. If Monte opens a door blindly, and it happens to be a goat by luck, then 50% is, indeed, correct. How does that fit into this visualization? This seems to be missing any insight into how Monte's door selection is involved.
ADDED: Legitimate probabilities can be up to 1/2 or 2/3, depending on Monte's decision rules.

 

[Ibix]

But there might be a detail missing here. If Monte opens a door blindly, and it happens to be a goat by luck, then 50% is, indeed, correct. How does that fit into this visualization? This seems to be missing any insight into how Monte's door selection is involved.

If Monty opens a door at random he could open the same random door for all three players. One player sees the car (and what happens next in this game decides what the overall outcome actually is), and of the other two 50% win and 50% lose. You might actually prefer nine games for this example - the car in the same place in all, and all nine combinations of door choice and accidental door open. Again, three contestants see the car, three win and three lose.

Monty Fall doesn't usually define what happens in the case where a car is revealed, although the possibility that it might have happened if he'd tripped differently is what drives the different probabilities.

 

[FactChecker]

Monty Fall doesn't usually define what happens in the case where a car is revealed, although the possibility that it might have happened if he'd tripped differently is what drives the different probabilities.

Suppose Monte makes his decision blindly and the game is declared invalid and restarted if he opens the prize door. Then, for valid games that continue, the probability of both remaining doors is 1/2.
Depending on what rules Monte follows, there are legitimate probabilities of 1/2 and 2/3. I don't see how that can be clearly addressed in your approach.

 

[Dale]

But there might be a detail missing here. If Monte opens a door blindly, and it happens to be a goat by luck, then 50% is, indeed, correct. How does that fit into this visualization?

It fits fine, but is not as compelling.

For the player that has the car behind their door, Monty cannot spoil the game and that player wins (since they are all sticking).

For the other 2 players there is a 1/2 probability each that Monty spoils their game. So, on average 2*1/2=1 is spoiled per round, and the remaining unspoiled player loses.

So of the unspoiled games there is indeed a 1/2 probability of winning and losing.

 

[DaveC426913]

This thread could have been two posts long. All we had to do was ask Facebook.

 

[Dale]

The probability of a Monty Hall thread being only 2 posts long is substantially less than 1/3.

 

[DaveC426913]

This thread could have been two posts long. All we had to do was ask Facebook.
View attachment 370118

I just realized this didn't land the way I thought it would. (especially since it's too low-rez to actually read the comments)
FBusers are usually dumb at best, outright ign'ant at worst.

But the comments in this Monty Hall posts were bang-on. Every one was "Always switch. This is old news." and "Switching doubles your odds from 1/3 to 2/3rds", etc.

 

[FactChecker]

FBusers are usually dumb at best, outright ign'ant at worst.

Hay!!! I take that personally. ..... well, ok.

But the comments in this Monty Hall posts were bang-on. Every one was "Always switch. This is old news." and "Switching doubles your odds from 1/3 to 2/3rds", etc.

If that was the FB consensus, I am surprised and impressed.

 

[DaveC426913]

If that was the FB consensus, I am surprised and impressed.

Yes. By a landslide. And not just in an intuitive sense; most seemed to know it probabilistically.

 

[DaveC426913]

 

[statdad]

If I understand your odd question correctly, you're asking what the first question of the "game" contributes?

a) If the first question is answered ("Door 1 sir"), in the idealized setting of the game your chance of being correct is 1/3. When the host opens a door it must be a door that doesn't hide the prize (according to the popular version of the game: there was no such rule in the TV show) so you STILL have a 1/3 chance of having the winning door. This means there is a 2/3 chance the prize is behind the remaining door, so it's better to switch.
b) Absent that first question (and if this isn't what you're questioning, my apologies). You don't get to ask to open a door: the host simply says
"One of these doors has a prize, behind it, the other two have goats (very nice goats, but not as nice as the real prize). Want to see a goat?" and one of the goat doors is opened. Now you have to choose one of the remaining doors. Now: originally, the probability of the goat being behind any particular door was 1/3. Now that one door has been removed without any input from you, the probability you pick the "winning door"* is 1/2.

There is no inherent contradiction here about which of 1/3 or 1/2 is correct since they come from two different games: one with a first question, one without.

* Unless you like goats as much as you like the proffered prize: then the probability of picking the winning door is 1.

 

[FactChecker]

When the host opens a door it must be a door that doesn't hide the prize (according to the popular version of the game: there was no such rule in the TV show)

Is that right? That is a critical difference. I didn't watch it enough to know if Monte ever opened a door with the prize. I was just told that he never did.

 

[Dale]

Is that right? That is a critical difference. I didn't watch it enough to know if Monte ever opened a door with the prize. I was just told that he never did.

He never did. It was not a stated rule, but we can infer that he had the knowledge where the car was and used that knowledge to open a different door.

 

[Ibix]

Is that right? That is a critical difference. I didn't watch it enough to know if Monte ever opened a door with the prize. I was just told that he never did.

AFAIK, the big difference between the problem and the gameshow is that Monty had other options besides just "open a door". The problem version of Monty is just an algorithm. The real one had options and could steer you away from the prize based on his reading of your likely reaction to what he offered.

Switching might actually be a bad strategy on the show if Monty knows you know some maths. Unless he knows you know he knows that and is double-bluffing you.

 

[FactChecker]

AFAIK, the big difference between the problem and the gameshow is that Monty had other options besides just "open a door". The problem version of Monty is just an algorithm. The real one had options and could steer you away from the prize based on his reading of your likely reaction to what he offered.

Switching might actually be a bad strategy on the show if Monty knows you know some maths. Unless he knows you know he knows that and is double-bluffing you.

If Monty is allowed to play games, that changes everything and brings game theory into the equation. In that case, there is no way to answer the problem without more information about his strategy.

 

[Roberto Pavani]

I'm amazed by the length of this thread. In a sense, it confirms what I've often thought about statistics and probability: the rules themselves are actually quite simple compared to many other areas of science, yet they are extremely counterintuitive.

The mistake is usually not in the mathematics. The mistake is in taking a mental shortcut and skipping one of the assumptions or conditioning steps because it "feels obvious".

For the Monty Hall problem, I've always found it easier to think about it in reverse:

• the probability that your initial choice is wrong is 2/3;
• Monty intentionally reveals a goat and does not change that probability;
• therefore, the remaining unopened door inherits that 2/3 probability.

So switching wins 2/3 of the time.

The amazing part is not the mathematics. The amazing part is how difficult it is for our intuition to accept such a simple result.

In that sense, it is almost more of a psychological test than a mathematical one.

I remember that the first time someone presented the problem to me, I had to think about it for quite a while. My intuition immediately said "50/50", but intuition is not enough here. Probability forces you to be rigorous and follow the rules carefully. If you follow the logic all the way through, the answer is to switch.

 

[FactChecker]

The amazing part is not the mathematics. The amazing part is how difficult it is for our intuition to accept such a simple result.

Conditional probability is tricky and full of counter-intuitive results.

Probability forces you to be rigorous and follow the rules carefully. If you follow the logic all the way through, the answer is to switch.

Exactly. And it can help you to understand where your initial intuition goes wrong. It's important to make Bayes' Rule your new intuition.

 

[PeroK]

One of the problems is that many people want to see probability as an abstract or intuitive notion of belief. Whereas, for a game like Monty Hall, it is a relative frequency.

In fact, in the genuinely controversial Sleeping Princess problem, this same dichotomy arises. With many people, including eminent thinkers, rejecting a purely computational, frequentist approach, in favour of a more woolly argument.

That's not to deny the role of the Bayesian approach in scenarios where the precise frequencies are not known.