8 ms·

There are a number of things I like about the monty hall problem. There's the history, the unintuitiveness, the subtle easy-to-screw-up nature of probability pr

There are a number of things I like about the monty hall problem. There's the history, the unintuitiveness, the subtle easy-to-screw-up nature of probability problems, the calculation, the sociology, and the overconfidence of wrong experts. Most of all, it's the calculation vs intuition that I like. You can do the calculation, or run simulations and prove correctness. In fact, it would be much harder to be so widely wrong now that any statistician can so easily just code up a computer to run it a zillion times and get an empirical answer. But despite that, your intuition might still say otherwise. I'm fascinated by the gap between intuition and logic particularly because it can be closed as soon as you find the right lens to take to a problem. I like the example in the article about 100 doors, or even a million doors, where it really helps drive it home in your gut. That moment when you find the right intuition for a thing is a magical one that makes me love math.

darig 5y ago
When I first heard of it in high school, I immediately coded a simulation and had an answer in less than 5 minutes that I could be confident about. I would have sat second-guessing myself for 5 minutes had I not done that.
paulcole 5y ago
My favorite thing is when somebody says, “what if I want to win the goat?” Because you instantly know they’re a contrarian for its own sake.
- deleted 5y ago
  [deleted]
Spivak 5y ago
I don’t think you need the 100 or 1,000,000 doors thing. If you decide before not to switch then your goal is to pick the car which you have a 1/3 chance to do. If you decide before to switch then your goal is to pick a goat which you have a 2/3 chance to do.
deleted 5y ago
[deleted]
dunham 5y ago
The thing that bothers me about this problem is that (as far as I recall) it's not asserted that the host always gives the player a chance to switch. If the host sometimes gives people a choice and sometimes doesn't, based on unknown criteria, then all bets are off.
- dllthomas 5y ago
  Yeah, for instance if the host only offers the switch when you've picked the car, obviously you shouldn't switch. That said, this does not seem to be the thread the respondents were pulling at...
- antonvs 5y ago
  In the probability puzzle, the chance to switch is part of the basis of the puzzle. It is always offered. In the actual game show, the rules were more flexible. The OP article mentions this. For example, the host could "offer the contestant cash NOT to switch." That introduces a more psychological poker-like aspect, but that example still allows for switching and doesn't change the underlying probabilities.
lachstar-x 5y ago
Can you help me understand - in my view the choice to switch doors or keep the same door is irrelevant, because even if you keep the same door you're making a choice that is now 2/3 of the right answer. If you switch or keep, you're still choosing from two doors that contain a car and a goat. The other door is no longer relevant and doesn't affect the new state at all. It's your perspective (narrowing the choice down) that changes, not the position of the car. Like getting a run of red in roulette and thinking the next one has gotta be black.
- ronyeh 5y ago
  Think from the perspective of your first choice. You had a 66.66% chance of choosing the wrong door. So now I just eliminate one of the doors you didn't choose. There are two left, including the one you chose first, which only had a 33.33% chance of being correct. Nothing else has changed about the problem. What if we started with a thousand doors? You chose 1, with a 999/1000 chance of being wrong. Of the remaining doors, I open 499 doors. The prize is behind one of the remaining 500 that you did not choose the first time. Do you want to have a chance to pick from one of the 500?
  - ziml77 5y ago
    I like expanding it out to 1000 doors. Even though the logic is the same between 3 and 1000 doors, the chances are more intuitive with more doors. I think the part that really throws the brain off with the 3 door version is that the host opening a door seems like another random event but it's actually dependent on the state of the game at that point. Our natural intuition of the chances is for a version of the game where you pick a door, then the host randomly picks (and does not open as you also haven't) a different door, and then you have to choose if you want to switch to the door that neither of you picked.
- TacticalCoder 5y ago
  > The other door is no longer relevant and doesn't affect the new state at all. There is no new state, that's the thing. By picking one door you split these in two groups: one group with 1/3 to win, the other with 2/3 to win. When the host reveals the door with a goat behind it from the group that is 2/3 to win, it doesn't change the fact that that group had 2/3 to win to begin with. I mean: that's how I see it.
mathrando 5y ago
Whether Monte Hall is counter intuitive is a function of how the question is phrased. When you phrase it in a way that underlines the mechanical nature of the host's decision, people get it right. When you phrase it in a way that suggests the host's choice is itself random, people get it wrong. I think the first formulation primes people to think of it from the perspective of the host, which is the right perspective for this problem.
- lnl 5y ago
  > When you phrase it in a way that suggests the host's choice is itself random, people get it wrong. In other words, they still get it right; they get it right for the separate question that that phrasing implies. If the host's choice is random, so that when you initially picked wrong it's equally probable that the host open the door with the car and then say "sorry, looks like you lost" (which is what I assumed when I first heard this problem, not being familiar with the show), then even if the host happened to open the door with a goat and give you a chance to switch, it doesn't matter if you take it or not. People are correct that, for that question, the probabilities are one in two for both of the remaining doors.
  - deleted 5y ago
    [deleted]
  - deleted 5y ago
    [deleted]
hn_throwaway_99 5y ago
In this case, though, I don't believe it's just calculation vs. intuition. The reason the 100 or million door problem helped me understand was because it clarified an implicit rule of the game: that the door the host chooses to open is not random. If it were random, it would make for a very boring game ("Uhh, there's the prize, you win, I guess that's game over.") Once I understood that the host would only ever open "goat" doors, that's when I better understood the argument.
- FabHK 5y ago
  Indeed, that is a crucial bit of information that is not made explicit in the original framing of the problem.
  - mkl 5y ago
    I think it's made pretty explicit. In this version: "Then, the host, who is well-aware of what’s going on behind the scenes, opens door #3, revealing one of the goats." The host has to open a door that doesn't show the car.
    - someguydave 5y ago
      Yeah - I don't think that statement clarifies. The rule that the host will never reveal the prize should be made explicit.
    - listenallyall 5y ago
      You're quoting Priceonomic's 2021 description of the problem. This is NOT the 1990 Parade Magazine description of the problem that generated all the responses.
      - mkl 5y ago
        The 1990 Parade Magazine description is almost identical (and actually more explicit, since "say #3" is a removable parenthetical): "the host, who knows what’s behind the doors, opens another door, say #3, which has a goat" -- https://web.archive.org/web/20130121183432/http://marilynvossavant.com/game-show-problem/ https://web.archive.org/web/20130121183432/http://marilynvos... The host's knowledge is explicitly mentioned, and the only purpose this could have is that so he can use it to avoid giving the game away.
2muchcoffeeman 5y ago
* Most of all, it's the calculation vs intuition that I like. You can do the calculation, or run simulations and prove correctness.* It puzzles me that so many smart people got it wrong. It’s so easy to check your working. When I first heard of this problem, I got it wrong too. Then I was told the answer, and to prove it to myself, it’s trivial to list the scenarios and simulate on paper. It’s still uncomfortable to think about sometimes, but I know how to show it’s true. But all these people never thought to just do the working? Crazy!
- mypalmike 5y ago
  I think people didn't check because the answer of 50% is so strikingly obvious (despite being wrong) that it seems like there's no work to check.
  - LanceH 5y ago
    It's right depending on your assumptions. If the host has rules of only picking the right most door the contestant doesn't pick, and it happens to be a goat then the answer is 50%. Basically if the host is just as willing to show a car as show a goat and it is a goat which is shown, the answer is 50%. There is a confirmation bias that people expect people who are aware of the Monty Hall Problem expect others to be wrong about it and attribute it to people not understanding. I think it's perfectly acceptable for someone to read the vague "the host, who is well-aware of what’s going on behind the scenes" and not assume that means he chooses to pick a goat. They're probably right, and the people saying 50% are reasoning along the lines of "there are two doors so 50/50". For rigor, I think the host's rules need to be more strict: "The host knows what is behind the doors and will never reveal the car". Or a 50% answer is correct if you lay out the principles of how the host acts given his knowledge.
    - fourthark 5y ago
      Intention doesn’t matter. The problem states that the host opens another door and reveals a goat. It doesn’t matter that the host meant to do that. Now that the information is revealed, the probabilities have changed. Obviously if the host revealed the car, that would also affect the probabilities!
kahirsch 5y ago
The day this was published in Parade magazine, I was sure that she was wrong. Even though I was a computer programmer then (1990), I didn't own a computer. I drove across town to my parent's house to use my father's computer to prover her wrong. I didn't even have to run the program. Just the act of writing the program made me realize that she was right. When I was writing the part where the host picks which door to open, I had a revelation.
- Talanes 5y ago
  Because in 2/3 scenarios you had to make the host choose the door without a car?
  - glitchc 5y ago
    The host can never choose the door with the car, because then the game is over. Ergo the host has to choose the door with goat, every time.
    - dekhn 5y ago
      this constraint was unclear to me in the original problem statement, which is why I thought she was wrong, too. It wasn't until somebody wrote in a letter saying they wrote a computer program that I understand what the rules were, and then it was obvious. I wish the problem was stated more clearly to say, the host has to choose the door with goat.
      - dvfjsdhgfv 5y ago
        > I wish the problem was stated more clearly to say, the host has to choose the door with goat. Yes, that one point would make many people instantly realize the probability changed. Nevertheless, you'd think mathematicians and statisticians sending her offensive letters should've known better.
    - Talanes 5y ago
      Right, and the host is only making a meaningful choice in the 2/3 scenarios where you didn't pick the car in the first place.
visarga 5y ago
This shows how much of our intelligence is crystalized (external). We look intelligent because we're reusing the solutions found by others. As soon as we can't do that it becomes apparent that the intelligence of a single person is not so great. It took us millennia to get where we are.
wildermuthn 5y ago
I was trying to think this through and realized it would be faster and more accurate to just write up a little script. Pretty sure this is accurate: https://replit.com/@NateWildermuth/Monty-Hall-Problem https://replit.com/@NateWildermuth/Monty-Hall-Problem Thinking about this was a lot easier once I could look at the code. It seems like what's happening is that when changing doors, you are actually changing to two doors rather than one, giving you 2/3 odds?

darig 5y ago

When I first heard of it in high school, I immediately coded a simulation and had an answer in less than 5 minutes that I could be confident about. I would have sat second-guessing myself for 5 minutes had I not done that.

paulcole 5y ago

My favorite thing is when somebody says, “what if I want to win the goat?” Because you instantly know they’re a contrarian for its own sake.

deleted 5y ago
[deleted]

Spivak 5y ago

I don’t think you need the 100 or 1,000,000 doors thing. If you decide before not to switch then your goal is to pick the car which you have a 1/3 chance to do. If you decide before to switch then your goal is to pick a goat which you have a 2/3 chance to do.

deleted 5y ago

[deleted]

dunham 5y ago

The thing that bothers me about this problem is that (as far as I recall) it's not asserted that the host always gives the player a chance to switch. If the host sometimes gives people a choice and sometimes doesn't, based on unknown criteria, then all bets are off.

dllthomas 5y ago
Yeah, for instance if the host only offers the switch when you've picked the car, obviously you shouldn't switch. That said, this does not seem to be the thread the respondents were pulling at...
antonvs 5y ago
In the probability puzzle, the chance to switch is part of the basis of the puzzle. It is always offered. In the actual game show, the rules were more flexible. The OP article mentions this. For example, the host could "offer the contestant cash NOT to switch." That introduces a more psychological poker-like aspect, but that example still allows for switching and doesn't change the underlying probabilities.

lachstar-x 5y ago

Can you help me understand - in my view the choice to switch doors or keep the same door is irrelevant, because even if you keep the same door you're making a choice that is now 2/3 of the right answer. If you switch or keep, you're still choosing from two doors that contain a car and a goat. The other door is no longer relevant and doesn't affect the new state at all. It's your perspective (narrowing the choice down) that changes, not the position of the car. Like getting a run of red in roulette and thinking the next one has gotta be black.

ronyeh 5y ago
Think from the perspective of your first choice. You had a 66.66% chance of choosing the wrong door. So now I just eliminate one of the doors you didn't choose. There are two left, including the one you chose first, which only had a 33.33% chance of being correct. Nothing else has changed about the problem. What if we started with a thousand doors? You chose 1, with a 999/1000 chance of being wrong. Of the remaining doors, I open 499 doors. The prize is behind one of the remaining 500 that you did not choose the first time. Do you want to have a chance to pick from one of the 500?
- ziml77 5y ago
  I like expanding it out to 1000 doors. Even though the logic is the same between 3 and 1000 doors, the chances are more intuitive with more doors. I think the part that really throws the brain off with the 3 door version is that the host opening a door seems like another random event but it's actually dependent on the state of the game at that point. Our natural intuition of the chances is for a version of the game where you pick a door, then the host randomly picks (and does not open as you also haven't) a different door, and then you have to choose if you want to switch to the door that neither of you picked.
TacticalCoder 5y ago
> The other door is no longer relevant and doesn't affect the new state at all. There is no new state, that's the thing. By picking one door you split these in two groups: one group with 1/3 to win, the other with 2/3 to win. When the host reveals the door with a goat behind it from the group that is 2/3 to win, it doesn't change the fact that that group had 2/3 to win to begin with. I mean: that's how I see it.

mathrando 5y ago

Whether Monte Hall is counter intuitive is a function of how the question is phrased. When you phrase it in a way that underlines the mechanical nature of the host's decision, people get it right. When you phrase it in a way that suggests the host's choice is itself random, people get it wrong. I think the first formulation primes people to think of it from the perspective of the host, which is the right perspective for this problem.

lnl 5y ago
> When you phrase it in a way that suggests the host's choice is itself random, people get it wrong. In other words, they still get it right; they get it right for the separate question that that phrasing implies. If the host's choice is random, so that when you initially picked wrong it's equally probable that the host open the door with the car and then say "sorry, looks like you lost" (which is what I assumed when I first heard this problem, not being familiar with the show), then even if the host happened to open the door with a goat and give you a chance to switch, it doesn't matter if you take it or not. People are correct that, for that question, the probabilities are one in two for both of the remaining doors.
- deleted 5y ago
  [deleted]
- deleted 5y ago
  [deleted]

hn_throwaway_99 5y ago

In this case, though, I don't believe it's just calculation vs. intuition. The reason the 100 or million door problem helped me understand was because it clarified an implicit rule of the game: that the door the host chooses to open is not random. If it were random, it would make for a very boring game ("Uhh, there's the prize, you win, I guess that's game over.") Once I understood that the host would only ever open "goat" doors, that's when I better understood the argument.

FabHK 5y ago
Indeed, that is a crucial bit of information that is not made explicit in the original framing of the problem.
- mkl 5y ago
  I think it's made pretty explicit. In this version: "Then, the host, who is well-aware of what’s going on behind the scenes, opens door #3, revealing one of the goats." The host has to open a door that doesn't show the car.
  - someguydave 5y ago
    Yeah - I don't think that statement clarifies. The rule that the host will never reveal the prize should be made explicit.
  - listenallyall 5y ago
    You're quoting Priceonomic's 2021 description of the problem. This is NOT the 1990 Parade Magazine description of the problem that generated all the responses.
    - mkl 5y ago
      The 1990 Parade Magazine description is almost identical (and actually more explicit, since "say #3" is a removable parenthetical): "the host, who knows what’s behind the doors, opens another door, say #3, which has a goat" -- https://web.archive.org/web/20130121183432/http://marilynvossavant.com/game-show-problem/ https://web.archive.org/web/20130121183432/http://marilynvos... The host's knowledge is explicitly mentioned, and the only purpose this could have is that so he can use it to avoid giving the game away.

2muchcoffeeman 5y ago

* Most of all, it's the calculation vs intuition that I like. You can do the calculation, or run simulations and prove correctness.* It puzzles me that so many smart people got it wrong. It’s so easy to check your working. When I first heard of this problem, I got it wrong too. Then I was told the answer, and to prove it to myself, it’s trivial to list the scenarios and simulate on paper. It’s still uncomfortable to think about sometimes, but I know how to show it’s true. But all these people never thought to just do the working? Crazy!

mypalmike 5y ago
I think people didn't check because the answer of 50% is so strikingly obvious (despite being wrong) that it seems like there's no work to check.
- LanceH 5y ago
  It's right depending on your assumptions. If the host has rules of only picking the right most door the contestant doesn't pick, and it happens to be a goat then the answer is 50%. Basically if the host is just as willing to show a car as show a goat and it is a goat which is shown, the answer is 50%. There is a confirmation bias that people expect people who are aware of the Monty Hall Problem expect others to be wrong about it and attribute it to people not understanding. I think it's perfectly acceptable for someone to read the vague "the host, who is well-aware of what’s going on behind the scenes" and not assume that means he chooses to pick a goat. They're probably right, and the people saying 50% are reasoning along the lines of "there are two doors so 50/50". For rigor, I think the host's rules need to be more strict: "The host knows what is behind the doors and will never reveal the car". Or a 50% answer is correct if you lay out the principles of how the host acts given his knowledge.
  - fourthark 5y ago
    Intention doesn’t matter. The problem states that the host opens another door and reveals a goat. It doesn’t matter that the host meant to do that. Now that the information is revealed, the probabilities have changed. Obviously if the host revealed the car, that would also affect the probabilities!

kahirsch 5y ago

The day this was published in Parade magazine, I was sure that she was wrong. Even though I was a computer programmer then (1990), I didn't own a computer. I drove across town to my parent's house to use my father's computer to prover her wrong. I didn't even have to run the program. Just the act of writing the program made me realize that she was right. When I was writing the part where the host picks which door to open, I had a revelation.

Talanes 5y ago
Because in 2/3 scenarios you had to make the host choose the door without a car?
- glitchc 5y ago
  The host can never choose the door with the car, because then the game is over. Ergo the host has to choose the door with goat, every time.
  - dekhn 5y ago
    this constraint was unclear to me in the original problem statement, which is why I thought she was wrong, too. It wasn't until somebody wrote in a letter saying they wrote a computer program that I understand what the rules were, and then it was obvious. I wish the problem was stated more clearly to say, the host has to choose the door with goat.
    - dvfjsdhgfv 5y ago
      > I wish the problem was stated more clearly to say, the host has to choose the door with goat. Yes, that one point would make many people instantly realize the probability changed. Nevertheless, you'd think mathematicians and statisticians sending her offensive letters should've known better.
  - Talanes 5y ago
    Right, and the host is only making a meaningful choice in the 2/3 scenarios where you didn't pick the car in the first place.

visarga 5y ago

This shows how much of our intelligence is crystalized (external). We look intelligent because we're reusing the solutions found by others. As soon as we can't do that it becomes apparent that the intelligence of a single person is not so great. It took us millennia to get where we are.

wildermuthn 5y ago

I was trying to think this through and realized it would be faster and more accurate to just write up a little script. Pretty sure this is accurate: https://replit.com/@NateWildermuth/Monty-Hall-Problem https://replit.com/@NateWildermuth/Monty-Hall-Problem Thinking about this was a lot easier once I could look at the code. It seems like what's happening is that when changing doors, you are actually changing to two doors rather than one, giving you 2/3 odds?