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My favorite example is Monty Hall problem. "Smart" people often use it as the evidence of how bad general people are at probability. It really isn't. The probl

My favorite example is Monty Hall problem. "Smart" people often use it as the evidence of how bad general people are at probability. It really isn't. The problem is usually given in this form: > 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. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, "Do you want to pick door No. 2?" Is it to your advantage to switch your choice? [1] The only correct ansewr to this question is "I don't know." People who answer yes are just taking way too many standardized tests. [1]: https://en.wikipedia.org/wiki/Monty_Hall_problem https://en.wikipedia.org/wiki/Monty_Hall_problem

copypasterepeat 2y ago
I guess I don't understand why you think "I don't know" is the only correct answer. It's clearly not, which is the point of the problem in the first place. It's a bit hard to grok, but once you do, it's clear what the right answer is.
- raincole 2y ago
  "I don't know" is the only correct anwser because the original problem isn't properly worded. You need to (at least) add this statement to the original question: > The host must open a door in any situation, and you know this rule. Because "host" in daily language is a human being with agency to choose whether to not open a door. Without this statement, the original problem is actually a game theory problem with two players. The host might be playing a strategy where he only opens the door when he knows you picked the car at the first place, or any other strategy. The above statement removes the ambiguity and that's what the original author meant (but failed to put in words). (If you're interesting in this, the wikipedia page linked above actually contains a quite extensive discussion in history over it)
  - dooglius 2y ago
    I don't think improper wording makes it an example of the type of puzzle GP is complaining about, because the counter-intuitive aspect holds up when everything is clarified.
  - copypasterepeat 2y ago
    Ah I see what you mean. Yeah, that makes sense. If the host behavior is not clearly stated then the right answer would indeed be "it can't be determined from available information".
- drdec 2y ago
  It depends on if Monty has the option to show another door or do nothing.
karmakurtisaani 2y ago
But it is in your advantage! Code a simulator and do some numerical experiments. The simulations where you switch end up winning more. Edit: from your other reply I see your beef is more with the wording. Edit edit: although I disagree with your point even then. But it seems to lead to rather fruitless argument, so let's leave it here.
- naniwaduni 2y ago
  I think reasoning with the 100-door variation is likely to be more enlightening than encoding your assumptions into a simulation.
  - karmakurtisaani 2y ago
    I'm sure it depends on the person. Some people like to see raw numbers, others a more general problem.
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LargeWu 2y ago
The difference here is that the Monty Hall problem has an explanation that while counterintuitive, is statistically sound. You should always switch, because the probability you picked the correct door is locked in at the time you made the choice between 3 doors. It is 1/3 that you picked correctly, and 2/3 that you picked incorrectly. The counterintuitive part is that if you switch, you are effectively selecting all the doors you did not pick originally. It's your original door, vs the field. The green hat problem hinges on subjective interpretations of the meaning of both "liar" and the different ways in which the liar's sentence may be false. It may be false because the liar owns many hats, none of which are green. Or they own many hats, only some of which are green. Or they own no hats. These are all reasonable interpretations of how the sentence might be false, and the answers presented are not necessarily mutually exclusive.
- hulium 2y ago
  Last time I claimed this was the correct answer, I was linked to the Monty Crawl problem. https://www.probability.ca/jeff/writing/montyfall.pdf https://www.probability.ca/jeff/writing/montyfall.pdf
  - LargeWu 2y ago
    That's fine, but given the parameters of the problem change with the Monty Crawl variant, it's not the same problem, and doesn't invalidate the answer of the base variant.
    - hulium 2y ago
      The problem is that the parameter was unspecified in the original problem. Your answer is equivalent to the shaky answer referred to in the paper. Without knowing that the host selects between 2 goats with 50/50 chance, you cannot give a general answer.
      - LargeWu 2y ago
        I don't follow, which parameter was unspecified in the original problem?
hansvm 2y ago
Perhaps there's another formulation you meant to write, but every time I've seen it, it's been equivalent to the thing you just asked, and the only potentially ambiguous part of the question is "to your advantage". If you stay then you have a 1/3 chance of getting a car, and if you switch you have a 2/3 chance. Yes, it's also correct that you don't "know" the result, and you might prefer goats to cars (even then you should probably sell the car and buy several goats), but there's a reasonable enough interpretation of "advantage" that you shouldn't dismiss the problem outright.
- raincole 2y ago
  I regret opening this can of worms called Monty Hall. But anyway I'll link the relevant section from Wikipedia: https://en.wikipedia.org/wiki/Monty_Hall_problem#Other_host_behaviors https://en.wikipedia.org/wiki/Monty_Hall_problem#Other_host_... > The version of the Monty Hall problem published in Parade in 1990 did not specifically state that the host would always open another door, or always offer a choice to switch, or even never open the door revealing the car. However, Savant made it clear in her second follow-up column that the intended host's behavior could only be what led to the 2/3 probability she gave as her original answer. (emphasis mine) My point was that when people ask this question, they often word it like the very 1990 version did, lefting out this critical statement (which Savant considered needed as well, therefore she clarified in the follow-up column), making the question ambiguous. (Although Savant also said "Very few (out of people who said is 2/3 is wrong) raised questions about ambiguity"... so perhaps people are actually just bad at probablity...?)
  - btilly 2y ago
    Just be glad that you didn't go for the 2 envelopes problem which was her next. Vos Savant was wrong about that one, and few people like the real answer. The problem is that you're presented with two envelopes with 2 real numbers inside. You randomly select one, then look, and try to guess if you got the larger number. It doesn't seem like you can do better than even, but you can! Unfortunately everyone hates the answer. Which is that you make up a random number and pretend it is the other one. Your odds of being right are 50% + (probability of choosing between the numbers) / 2 Which can always be strictly bigger than 50%. (Though possibly by only a little amount.) Special case. If those numbers and yours were all independently randomly chosen from the same distribution, you'll be right 2/3 of the time.
quchen 2y ago
Can you explain the mistake (of people taking the standardized tests)? I don't understand it from the message, and the Wikipedia article seems to detail exactly the "normal, smart" switching solution.
suggestion 2y ago
Perhaps it'll be more intuitive to you if you scale the number of doors up. If there are 100 doors, only one containing a car, you pick one, the host reveals 98/99 remaining doors as goats, it's obvious the correct choice is to switch. The correct answer is a mathematically provable probability. 1% chance you picked the right door, 99% chance the door was in the remaining pool, therefore 99% chance the last remaining door is the correct door.
- oneshtein 2y ago
  If you will try to play this game in real life, with such logic it will be easy to trick you 100% of the time.
  - suggestion 2y ago
    You can write a simulation that does this (as MIT and several others have) and Monte Carlo it. You will find that the logic is 100% correct. The prize MUST be either the door you picked initially or the one you can switch to. There is a 99% chance it's not the one you picked. https://web.mit.edu/rsi/www/2013/files/MiniSamples/MontyHall/montymain.pdf https://web.mit.edu/rsi/www/2013/files/MiniSamples/MontyHall...

copypasterepeat 2y ago

I guess I don't understand why you think "I don't know" is the only correct answer. It's clearly not, which is the point of the problem in the first place. It's a bit hard to grok, but once you do, it's clear what the right answer is.

raincole 2y ago
"I don't know" is the only correct anwser because the original problem isn't properly worded. You need to (at least) add this statement to the original question: > The host must open a door in any situation, and you know this rule. Because "host" in daily language is a human being with agency to choose whether to not open a door. Without this statement, the original problem is actually a game theory problem with two players. The host might be playing a strategy where he only opens the door when he knows you picked the car at the first place, or any other strategy. The above statement removes the ambiguity and that's what the original author meant (but failed to put in words). (If you're interesting in this, the wikipedia page linked above actually contains a quite extensive discussion in history over it)
- dooglius 2y ago
  I don't think improper wording makes it an example of the type of puzzle GP is complaining about, because the counter-intuitive aspect holds up when everything is clarified.
- copypasterepeat 2y ago
  Ah I see what you mean. Yeah, that makes sense. If the host behavior is not clearly stated then the right answer would indeed be "it can't be determined from available information".
drdec 2y ago
It depends on if Monty has the option to show another door or do nothing.

karmakurtisaani 2y ago

But it is in your advantage! Code a simulator and do some numerical experiments. The simulations where you switch end up winning more. Edit: from your other reply I see your beef is more with the wording. Edit edit: although I disagree with your point even then. But it seems to lead to rather fruitless argument, so let's leave it here.

naniwaduni 2y ago
I think reasoning with the 100-door variation is likely to be more enlightening than encoding your assumptions into a simulation.
- karmakurtisaani 2y ago
  I'm sure it depends on the person. Some people like to see raw numbers, others a more general problem.

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LargeWu 2y ago

The difference here is that the Monty Hall problem has an explanation that while counterintuitive, is statistically sound. You should always switch, because the probability you picked the correct door is locked in at the time you made the choice between 3 doors. It is 1/3 that you picked correctly, and 2/3 that you picked incorrectly. The counterintuitive part is that if you switch, you are effectively selecting all the doors you did not pick originally. It's your original door, vs the field. The green hat problem hinges on subjective interpretations of the meaning of both "liar" and the different ways in which the liar's sentence may be false. It may be false because the liar owns many hats, none of which are green. Or they own many hats, only some of which are green. Or they own no hats. These are all reasonable interpretations of how the sentence might be false, and the answers presented are not necessarily mutually exclusive.

hulium 2y ago
Last time I claimed this was the correct answer, I was linked to the Monty Crawl problem. https://www.probability.ca/jeff/writing/montyfall.pdf https://www.probability.ca/jeff/writing/montyfall.pdf
- LargeWu 2y ago
  That's fine, but given the parameters of the problem change with the Monty Crawl variant, it's not the same problem, and doesn't invalidate the answer of the base variant.
  - hulium 2y ago
    The problem is that the parameter was unspecified in the original problem. Your answer is equivalent to the shaky answer referred to in the paper. Without knowing that the host selects between 2 goats with 50/50 chance, you cannot give a general answer.
    - LargeWu 2y ago
      I don't follow, which parameter was unspecified in the original problem?

hansvm 2y ago

Perhaps there's another formulation you meant to write, but every time I've seen it, it's been equivalent to the thing you just asked, and the only potentially ambiguous part of the question is "to your advantage". If you stay then you have a 1/3 chance of getting a car, and if you switch you have a 2/3 chance. Yes, it's also correct that you don't "know" the result, and you might prefer goats to cars (even then you should probably sell the car and buy several goats), but there's a reasonable enough interpretation of "advantage" that you shouldn't dismiss the problem outright.

raincole 2y ago
I regret opening this can of worms called Monty Hall. But anyway I'll link the relevant section from Wikipedia: https://en.wikipedia.org/wiki/Monty_Hall_problem#Other_host_behaviors https://en.wikipedia.org/wiki/Monty_Hall_problem#Other_host_... > The version of the Monty Hall problem published in Parade in 1990 did not specifically state that the host would always open another door, or always offer a choice to switch, or even never open the door revealing the car. However, Savant made it clear in her second follow-up column that the intended host's behavior could only be what led to the 2/3 probability she gave as her original answer. (emphasis mine) My point was that when people ask this question, they often word it like the very 1990 version did, lefting out this critical statement (which Savant considered needed as well, therefore she clarified in the follow-up column), making the question ambiguous. (Although Savant also said "Very few (out of people who said is 2/3 is wrong) raised questions about ambiguity"... so perhaps people are actually just bad at probablity...?)
- btilly 2y ago
  Just be glad that you didn't go for the 2 envelopes problem which was her next. Vos Savant was wrong about that one, and few people like the real answer. The problem is that you're presented with two envelopes with 2 real numbers inside. You randomly select one, then look, and try to guess if you got the larger number. It doesn't seem like you can do better than even, but you can! Unfortunately everyone hates the answer. Which is that you make up a random number and pretend it is the other one. Your odds of being right are 50% + (probability of choosing between the numbers) / 2 Which can always be strictly bigger than 50%. (Though possibly by only a little amount.) Special case. If those numbers and yours were all independently randomly chosen from the same distribution, you'll be right 2/3 of the time.

quchen 2y ago

Can you explain the mistake (of people taking the standardized tests)? I don't understand it from the message, and the Wikipedia article seems to detail exactly the "normal, smart" switching solution.

suggestion 2y ago

Perhaps it'll be more intuitive to you if you scale the number of doors up. If there are 100 doors, only one containing a car, you pick one, the host reveals 98/99 remaining doors as goats, it's obvious the correct choice is to switch. The correct answer is a mathematically provable probability. 1% chance you picked the right door, 99% chance the door was in the remaining pool, therefore 99% chance the last remaining door is the correct door.

oneshtein 2y ago
If you will try to play this game in real life, with such logic it will be easy to trick you 100% of the time.
- suggestion 2y ago
  You can write a simulation that does this (as MIT and several others have) and Monte Carlo it. You will find that the logic is 100% correct. The prize MUST be either the door you picked initially or the one you can switch to. There is a 99% chance it's not the one you picked. https://web.mit.edu/rsi/www/2013/files/MiniSamples/MontyHall/montymain.pdf https://web.mit.edu/rsi/www/2013/files/MiniSamples/MontyHall...