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> Why would you test it? I recall conversations on Usenet decades ago about the Monty Hall problem[1] in which people gave elementary proofs that probabilities

> Why would you test it? I recall conversations on Usenet decades ago about the Monty Hall problem[1] in which people gave elementary proofs that probabilities don't change by opening a door. Even from mathematicians and statisticians. People were very insistent that the analytical solution was simple and obvious and that switching doors didn't change anything. The only thing that changed some people's minds was a program that simulated the Monty Hall problem. This was needed to get people to reconsider their proof when the claim was highly counterintuitive. [1] https://en.wikipedia.org/wiki/Monty_Hall_problem https://en.wikipedia.org/wiki/Monty_Hall_problem

I_Am_Nous 3y ago
The Monty Hall Problem is a fun one because you can try to approach it from a purely analytical perspective and get one answer, while incorporating the whole situation (especially the fact that the final probability is not natural as they force the final decision into far fewer doors than originally present) and testing you can find a different answer entirely. I suppose this is an interesting corollary with discoveries made by deep theoretical mathematics. While something may seem possible because "the math checks out" it could be only theoretically possible as it relies on some unnatural value to "be" possible in the first place. Testing is where hopeful theories are smashed by reality until all that remains is the verifiable truth. Truly, why wouldn't we test?
- pmontra 3y ago
  That's how it went when I was solving problems at the Statistics course at university. I modeled the problem perfectly, got the wrong result. Changed assumptions, got the wrong result. Checked the solution, its reasoning didn't make much sense anyway. Run a simulation, got an approximate result close to the correct solution.
  - dotancohen 3y ago
    This sounds like the classic "tweak the model until the results fit with our preexisting conclusion". Very common across all industries unfortunately.
    - zh3 3y ago
      Also known in a derogatory fashion [0] as "adding epicycles" (after the Ptolemaic view of the heavens). [0] https://en.wikipedia.org/wiki/Deferent_and_epicycle#Bad_science https://en.wikipedia.org/wiki/Deferent_and_epicycle#Bad_scie...
- naniwaduni 3y ago
  Fundamentally, the trouble with the Monty Hall problem isn't that analysis comes to the wrong answer, it's that people often come to the wrong model when reasoning about it informally. It's not any harder to do the "correct" analysis than to write up a simulation. It's mostly just easier to convince yourself that the simulation matches the problem description when it reaches the unintuitive result.
  - s1artibartfast 3y ago
    Fundamentally, I think the real trouble with the Monty Hall problem is that the assumptions of the game are not clearly stated. Because of this, people come up with different models.
    - Natsu 3y ago
      This modelling ambiguity is resolved by do calculus, which makes a clear distinction between intervention and observation: https://arxiv.org/pdf/1305.5506.pdf https://arxiv.org/pdf/1305.5506.pdf
    - alexdowad 3y ago
      That's absolutely right; further, if you explicitly model the behavior of the game show host, you can exhibit models under which "it's better to switch" and models under which "it doesn't matter if you switch or not".
      - vince3455 3y ago
        >models under which "it doesn't matter if you switch or not". Could you provide an example? It seems obvious that a switcher wins exactly when a non switcher looses, which is 2 / 3 ?
        alexdowad 3y ago
        Take a game show host who lets you choose a door, randomly reveals what is behind one other door, and then gives you an opportunity to change your choice. This game show host CAN (randomly) reveal the prize; he has equal probability of revealing ANY of the unchosen doors. Say you are playing the Monty Hall game with this host. You choose your door, he opens another door, and it happens (purely by chance) that there is no prize there. Do you still believe that you have a 2/3 chance of winning if you switch to the other unopened door?
CuriouslyC 3y ago
You can demonstrate the Monty Hall problem solution analytically with Bayesian statistics using prior probabilities, no need to go all the way to Monte Carlo methods.
- TOMDM 3y ago
  As a proof the math is entirely sufficient, but for those who may struggle with it the simulation is persuasive.
- selimthegrim 3y ago
  This was how I convinced someone (RIP) by really stressing every element in the definition of Bayes’ theorem and probability space.
chankstein38 3y ago
This was how I built an intuition into the Monty Hall problem as well! Wrote a little app that simulated it a decade or so ago when I was discussing with friends!
deleted 3y ago
[deleted]
dfxm12 3y ago
The Monty Hall problem is especially unintuitive if you've ever watched Let's Make a Deal, since the problem set up is oh so close to, but not exactly, the set up of the Big Deal in the show. It's too easy to conflate the rules of the show with the math problem, which will lead to confusion. I think seeing the results of a simulation also elucidates the set up of the math problem vs reading a proof.
- cortesoft 3y ago
  Yeah, I feel like the Monty Hall confusion goes away if you are explicit about the rules: "Hall will always open one of the two non-chosen doors and will never reveal the prize" I think most people who don't understand the problem miss that critical detail.
  - bigstrat2003 3y ago
    No, I don't think that detail makes it any easier. I know that but I still really can't accept the correctness of the Monty Hall strategy (I have to basically just take it on faith and stop trying to understand it). I was trying to put my finger on why, and I think it's this. After Monty eliminates one of the three doors, then the prize is behind one of the two. If someone were to come in this point, with no prior knowledge whatsoever, their chance of picking the correct door at random is 1/2. And that is still true even if they pick the door which our contestant is being asked whether or not to switch from! This is a real mind fuck to try to accept, that the same state of what's behind each door leads to different odds of making a correct random choice, depending on when you make the choice. I honestly don't think I'll ever be able to "get" the Monty Hall strategy. I think I get why it works (choosing to switch means you're going from a 1/3 probability to 1/2), but it makes no sense at all. It seems like even if you choose to stay on the same door, your probability is 1/2 (the same as if Joe came in off the street and chose the same door as you). Like I said, I just have to take it on faith.
    - benchaney 3y ago
      The best probability estimate you can make is constrained by the information you have available. The new person showing up has less information than the existing constant, so it makes sense that their best estimate would be less precise. Similarly, if someone with x-ray vision walked up in the middle of the game, they could pick the car 100% of the time, because they have access to more information than either of the existing contestants. Your last paragraph isn't correct though, By switching you go from a 1/3 probability to a 2/3 probability. Based on the information the original contestant has, switching gets the car 2/3 of the time.
4star3star 3y ago
The way the problem makes sense to me is this. Doors: Goat Goat Car You pick a door. Monty shows you a Goat. You switch or stay. Monty will never show you the Car before offering a switch. He always shows you a Goat. It doesn't matter which Goat he shows you - it's just "not the Car". If your first choice is a Goat, switching will win you the Car. If your first choice is a Car, switching will win you a Goat. You have a 2/3 chance of picking a Goat, so, effectively, you want to pick a Goat so that you switch to the Car.
- tacitusarc 3y ago
  For sufficiently analytical folks that works, but for lay people it tends to still be confusing. The best way I’ve heard it explained to help people get it through intuition is by changing the number of doors and goats. Say there are 100 doors, and they all have goats except one, which has a car. You pick door 1. Monty then proceeds to open doors 2 through 48, skips door 49, and then opens the remaining doors. After all that, he stops and asks you, would you like to switch?
  - thaumasiotes 3y ago
    There's a better way to think about it. 1. You pick a door. 2. You get the offer "Do you want to keep that door, or choose both [all] of the other doors? In either case, you'll keep anything that isn't a goat." 3. Nobody opens any doors. Should you keep your one door, or switch to the two doors?
    - tomrod 3y ago
      This is the right way to think about it. Very clear and concise, thank you.
  - matsemann 3y ago
    I've never been happy with that explanation. I don't get why the host would not just open a single door, that's what the host does in the other scenario to me.
jackfoxy 3y ago
The Monty Hall problem is a fun one to code up, and yeah, there are otherwise smart people who refuse to believe it. I coded it up in F# https://github.com/jackfoxy/LetsMakeADeal https://github.com/jackfoxy/LetsMakeADeal to convince one of the founders of a start-up I worked for. He just grunted and walked away. Pretty sure he still doesn't want to hear about Bayes' Theorem.
- mensetmanusman 3y ago
  Just start with an infinite number of doors and move backwards from that.
bscphil 3y ago
I think the great advantage of "simulation", for the programming-literate, is not that you can simulate your way to a correct answer, but that the process of creating a simulation is likely to show you the error in your reasoning. As a young teenager, I encountered the Monty Hall problem for the first time, and I didn't believe that the "analytical" answer was correct. I decided to simulate it by programming. In the 20 minutes it took me to write a simulation, I went from complete incomprehension to a full understanding of why I got the results I got. Programming a simulation of the problem forces you to write out the algorithmic significance of "Monty reveals one of the goats".
strangattractor 3y ago
The Monty Hall problem is small enough to enumerate all the out comes on paper. If you then test all the different scenarios (not many) and compare switching to not switching - switching produces a slight edge. So it can be done without a computer and not a great deal of effort. I am not a mathematician so it was the only way I could prove it to myself at the time.
lcnPylGDnU4H9OF 3y ago
Something that makes this a lot more intuitive is to increase the number of doors. If there's 100 doors, the host will open all remaining doors except 1, and they will never open the door with the car behind it, then one has a 1% chance of winning the car if they don't switch doors and it would happen only because they initially chose the door with the car.
tshaddox 3y ago
I'm curious. When those people see the simulation, do they then go back to the analysis and uncover their mistaken reasoning? Or do they just continue to reject the analysis but begrudgingly accept the outcome of the simulation? The analysis of the Monty Hall problem is so very simple I find it very odd to staunchly reject it but then be persuaded by the simulation.
- cantrevealname 3y ago
  > is so very simple Years ago, when the Monty Hall problem was not well known, I've seen with my own eyes that it was hard to convince some very smart people of the correct answer. Indeed, after being convinced through simulation or exhaustive enumeration of the decision tree or some other way, they would go back and see what went wrong with their initial analysis.
Natsu 3y ago
This isn't so mysterious once you learn a bit of do calculus and realize that observation and intervention are fundamentally different things.

I_Am_Nous 3y ago

The Monty Hall Problem is a fun one because you can try to approach it from a purely analytical perspective and get one answer, while incorporating the whole situation (especially the fact that the final probability is not natural as they force the final decision into far fewer doors than originally present) and testing you can find a different answer entirely. I suppose this is an interesting corollary with discoveries made by deep theoretical mathematics. While something may seem possible because "the math checks out" it could be only theoretically possible as it relies on some unnatural value to "be" possible in the first place. Testing is where hopeful theories are smashed by reality until all that remains is the verifiable truth. Truly, why wouldn't we test?

pmontra 3y ago
That's how it went when I was solving problems at the Statistics course at university. I modeled the problem perfectly, got the wrong result. Changed assumptions, got the wrong result. Checked the solution, its reasoning didn't make much sense anyway. Run a simulation, got an approximate result close to the correct solution.
- dotancohen 3y ago
  This sounds like the classic "tweak the model until the results fit with our preexisting conclusion". Very common across all industries unfortunately.
  - zh3 3y ago
    Also known in a derogatory fashion [0] as "adding epicycles" (after the Ptolemaic view of the heavens). [0] https://en.wikipedia.org/wiki/Deferent_and_epicycle#Bad_science https://en.wikipedia.org/wiki/Deferent_and_epicycle#Bad_scie...
naniwaduni 3y ago
Fundamentally, the trouble with the Monty Hall problem isn't that analysis comes to the wrong answer, it's that people often come to the wrong model when reasoning about it informally. It's not any harder to do the "correct" analysis than to write up a simulation. It's mostly just easier to convince yourself that the simulation matches the problem description when it reaches the unintuitive result.
- s1artibartfast 3y ago
  Fundamentally, I think the real trouble with the Monty Hall problem is that the assumptions of the game are not clearly stated. Because of this, people come up with different models.
  - Natsu 3y ago
    This modelling ambiguity is resolved by do calculus, which makes a clear distinction between intervention and observation: https://arxiv.org/pdf/1305.5506.pdf https://arxiv.org/pdf/1305.5506.pdf
  - alexdowad 3y ago
    That's absolutely right; further, if you explicitly model the behavior of the game show host, you can exhibit models under which "it's better to switch" and models under which "it doesn't matter if you switch or not".
    - vince3455 3y ago
      >models under which "it doesn't matter if you switch or not". Could you provide an example? It seems obvious that a switcher wins exactly when a non switcher looses, which is 2 / 3 ?
      - alexdowad 3y ago
        Take a game show host who lets you choose a door, randomly reveals what is behind one other door, and then gives you an opportunity to change your choice. This game show host CAN (randomly) reveal the prize; he has equal probability of revealing ANY of the unchosen doors. Say you are playing the Monty Hall game with this host. You choose your door, he opens another door, and it happens (purely by chance) that there is no prize there. Do you still believe that you have a 2/3 chance of winning if you switch to the other unopened door?

CuriouslyC 3y ago

You can demonstrate the Monty Hall problem solution analytically with Bayesian statistics using prior probabilities, no need to go all the way to Monte Carlo methods.

TOMDM 3y ago
As a proof the math is entirely sufficient, but for those who may struggle with it the simulation is persuasive.
selimthegrim 3y ago
This was how I convinced someone (RIP) by really stressing every element in the definition of Bayes’ theorem and probability space.

chankstein38 3y ago

This was how I built an intuition into the Monty Hall problem as well! Wrote a little app that simulated it a decade or so ago when I was discussing with friends!

deleted 3y ago

[deleted]

dfxm12 3y ago

The Monty Hall problem is especially unintuitive if you've ever watched Let's Make a Deal, since the problem set up is oh so close to, but not exactly, the set up of the Big Deal in the show. It's too easy to conflate the rules of the show with the math problem, which will lead to confusion. I think seeing the results of a simulation also elucidates the set up of the math problem vs reading a proof.

cortesoft 3y ago
Yeah, I feel like the Monty Hall confusion goes away if you are explicit about the rules: "Hall will always open one of the two non-chosen doors and will never reveal the prize" I think most people who don't understand the problem miss that critical detail.
- bigstrat2003 3y ago
  No, I don't think that detail makes it any easier. I know that but I still really can't accept the correctness of the Monty Hall strategy (I have to basically just take it on faith and stop trying to understand it). I was trying to put my finger on why, and I think it's this. After Monty eliminates one of the three doors, then the prize is behind one of the two. If someone were to come in this point, with no prior knowledge whatsoever, their chance of picking the correct door at random is 1/2. And that is still true even if they pick the door which our contestant is being asked whether or not to switch from! This is a real mind fuck to try to accept, that the same state of what's behind each door leads to different odds of making a correct random choice, depending on when you make the choice. I honestly don't think I'll ever be able to "get" the Monty Hall strategy. I think I get why it works (choosing to switch means you're going from a 1/3 probability to 1/2), but it makes no sense at all. It seems like even if you choose to stay on the same door, your probability is 1/2 (the same as if Joe came in off the street and chose the same door as you). Like I said, I just have to take it on faith.
  - benchaney 3y ago
    The best probability estimate you can make is constrained by the information you have available. The new person showing up has less information than the existing constant, so it makes sense that their best estimate would be less precise. Similarly, if someone with x-ray vision walked up in the middle of the game, they could pick the car 100% of the time, because they have access to more information than either of the existing contestants. Your last paragraph isn't correct though, By switching you go from a 1/3 probability to a 2/3 probability. Based on the information the original contestant has, switching gets the car 2/3 of the time.

4star3star 3y ago

The way the problem makes sense to me is this. Doors: Goat Goat Car You pick a door. Monty shows you a Goat. You switch or stay. Monty will never show you the Car before offering a switch. He always shows you a Goat. It doesn't matter which Goat he shows you - it's just "not the Car". If your first choice is a Goat, switching will win you the Car. If your first choice is a Car, switching will win you a Goat. You have a 2/3 chance of picking a Goat, so, effectively, you want to pick a Goat so that you switch to the Car.

tacitusarc 3y ago
For sufficiently analytical folks that works, but for lay people it tends to still be confusing. The best way I’ve heard it explained to help people get it through intuition is by changing the number of doors and goats. Say there are 100 doors, and they all have goats except one, which has a car. You pick door 1. Monty then proceeds to open doors 2 through 48, skips door 49, and then opens the remaining doors. After all that, he stops and asks you, would you like to switch?
- thaumasiotes 3y ago
  There's a better way to think about it. 1. You pick a door. 2. You get the offer "Do you want to keep that door, or choose both [all] of the other doors? In either case, you'll keep anything that isn't a goat." 3. Nobody opens any doors. Should you keep your one door, or switch to the two doors?
  - tomrod 3y ago
    This is the right way to think about it. Very clear and concise, thank you.
- matsemann 3y ago
  I've never been happy with that explanation. I don't get why the host would not just open a single door, that's what the host does in the other scenario to me.

jackfoxy 3y ago

The Monty Hall problem is a fun one to code up, and yeah, there are otherwise smart people who refuse to believe it. I coded it up in F# https://github.com/jackfoxy/LetsMakeADeal https://github.com/jackfoxy/LetsMakeADeal to convince one of the founders of a start-up I worked for. He just grunted and walked away. Pretty sure he still doesn't want to hear about Bayes' Theorem.

mensetmanusman 3y ago
Just start with an infinite number of doors and move backwards from that.

bscphil 3y ago

I think the great advantage of "simulation", for the programming-literate, is not that you can simulate your way to a correct answer, but that the process of creating a simulation is likely to show you the error in your reasoning. As a young teenager, I encountered the Monty Hall problem for the first time, and I didn't believe that the "analytical" answer was correct. I decided to simulate it by programming. In the 20 minutes it took me to write a simulation, I went from complete incomprehension to a full understanding of why I got the results I got. Programming a simulation of the problem forces you to write out the algorithmic significance of "Monty reveals one of the goats".

strangattractor 3y ago

The Monty Hall problem is small enough to enumerate all the out comes on paper. If you then test all the different scenarios (not many) and compare switching to not switching - switching produces a slight edge. So it can be done without a computer and not a great deal of effort. I am not a mathematician so it was the only way I could prove it to myself at the time.

lcnPylGDnU4H9OF 3y ago

Something that makes this a lot more intuitive is to increase the number of doors. If there's 100 doors, the host will open all remaining doors except 1, and they will never open the door with the car behind it, then one has a 1% chance of winning the car if they don't switch doors and it would happen only because they initially chose the door with the car.

tshaddox 3y ago

I'm curious. When those people see the simulation, do they then go back to the analysis and uncover their mistaken reasoning? Or do they just continue to reject the analysis but begrudgingly accept the outcome of the simulation? The analysis of the Monty Hall problem is so very simple I find it very odd to staunchly reject it but then be persuaded by the simulation.

cantrevealname 3y ago
> is so very simple Years ago, when the Monty Hall problem was not well known, I've seen with my own eyes that it was hard to convince some very smart people of the correct answer. Indeed, after being convinced through simulation or exhaustive enumeration of the decision tree or some other way, they would go back and see what went wrong with their initial analysis.

Natsu 3y ago

This isn't so mysterious once you learn a bit of do calculus and realize that observation and intervention are fundamentally different things.