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It would be interesting to study whether programmers reading a simulation of monty-hall would predict the same outcomes from the english-language question. My

It would be interesting to study whether programmers reading a simulation of monty-hall would predict the same outcomes from the english-language question. My view of many of these "non-intuitive" probability questions is the terrible phrasing favoured by the sorts who pose these problems. I'm fundamentally sceptical that animals have, in general, a "poor" understanding of probability. I think it vastly more likely that natural language is a highly lossy communication system ill-fit for posing abstract problems that are outside of our ecological intuitions. It isn't probability at issue, but correctly recovering exactly what problem we're supposed to be considering. Writing a program forces all assumptions into the open, and I'd bet you'd see far fewer mistakes -- yet our so-called intuitions about probability would be the same.

vitus 2y ago
> I'm fundamentally sceptical that animals have, in general, a "poor" understanding of probability. I take the opposite view. Casinos with all of their negative-expected-value (for players) games would not exist, nor would many lotteries, if not for the allure of "oh, but if I win big..." overpowering any rational decision-making. Monty Hall is not challenging as a result of information loss due to translating to natural language. Reading the simulation code is like telling ChatGPT to think about the problem step-by-step: if you picked the door with the car, the host opens one of the goat doors arbitrarily; if you pick a door with a goat, the host opens the other goat door. Further, if you flip the sequence of events, such that the host opens a door containing a goat, and then you pick, you end up with a different probability of success entirely -- 50%, instead of 66.7% (always switch) or 33.3% (never switch)! This played out in the actual game show in real time, where people could observe the actual events directly. Per Monty Hall himself: > He said he was not surprised at the experts' insistence that the probability was 1 out of 2. "That's the same assumption contestants would make on the show after I showed them there was nothing behind one door," he said. "They'd think the odds on their door had now gone up to 1 in 2, so they hated to give up the door no matter how much money I offered." https://en.wikipedia.org/wiki/Monty_Hall_problem#History https://en.wikipedia.org/wiki/Monty_Hall_problem#History
- mjburgess 2y ago
  I'm not sure you've said anything that disagrees with me. My claim is that the information in the environment that people are sensitive to causes them to fail to accurately model the scenario. The problem they are solving, in their heads, fails to correspond to reality. My claim is: if it did, then they would solve it correctly. This is often phrased, typically by self-important mathematicians, that people's understanding of probability is wrong. This is a very bizarre view, and rather self-aggrandizing as far as probability goes. Probability isn't understanding or formulating the problem. Probability is merely the machinery, very basic, which runs upon a formulation. Do this instead: run one round of a game with 10 doors, and have people observe Monty opening all the relevant doors. Why should it be that if people observe this it becomes obvious, and yet if put in natural language it isn't? Natural language applied to non-intutive ecologies (ie., environments outside those we usually operate in) cause people to fail to model the sitaution correctly. Take casinos: when animals go out foraging they are playing a slot machine. Their addiction to foraging is perfectly ecologically rational. Casions exploit an information asymmetry: people cannot see the long-run house advantage. This has nothing to do with people's "probability" skills, and just their ability to accurately understand the sitatuon. The underlying foolishness behind those who equivocate between "probability" and "modelling scenarios" is the assumption that reality itself has probabilities, and that scenarios are easy to model. It is incredibly difficult to come up with an accurate cognitive model of wtf is going on in a monty hall game. So much so that famous mathematicians, with arbitrarily good probability skills, get the answer wrong. Teaching people "probability" here is beside the point. Animals are acutely good at it when they are presented with an ecologically intutive problem, of arbitary probablistic complexity.
  - vitus 2y ago
    Ah. I had focused on the lossiness of natural language much more than the rest of your statement. > Why should it be that if people observe this it becomes obvious, and yet if put in natural language it isn't? It's not clear to me that the general public would come to that conclusion, as opposed to the obvious extension of Hall's observation where contestants thought their odds having gone up upon being shown additional information. To you and me, yes, it might be even more obvious because we're familiar with the core principles. > Take casinos: when animals go out foraging they are playing a slot machine. Their addiction to foraging is perfectly ecologically rational. I'm inclined to disagree here. Surely animals don't just engage on a random walk; they know where they've found berries before, or can smell prey from a distance. And describing it as an addiction feels wrong. Once the animals are full, they don't keep hunting for food at the expense of other life-sustaining activities. > Casinos exploit an information asymmetry: people cannot see the long-run house advantage. Sure. I have also encountered hubris in the form of people who are aware of the house's edge but are confident they can do better than everyone else. There are also betting strategies that have positive expected value (e.g. lose a coin flip? double your bet; win a coin flip? reset to $1) but fail to take into consideration that you have probability = 1 of going bankrupt in the long run, since you don't have infinite money. > The underlying foolishness behind those who equivocate between "probability" and "modelling scenarios" is the assumption that reality itself has probabilities, and that scenarios are easy to model. In many cases, we use probabilities to approximate unknowns. Efficacy of a vaccine? Prevents infection X% of the time, or severe illness Y% of the time. Why not 100%? We don't necessarily know (or maybe it's due to a bunch of complex factors). But we've observed these outcomes for these samples of our population, and reducing it to a single number is "good enough".
kqr 2y ago
> I'm fundamentally sceptical that animals have, in general, a "poor" understanding of probability. I agree with this skepticism, and I think it boils more down to insufficient statistical literacy. Statistical statements (what things are actually like) often stand in contrast to logical reasoning (what things ought to be like) – popularly observed in quips such as "in theory there is no difference between theory and practice." This difference is uncomfortable and so many people (especially those that are better educated and well versed in logical reasoning – e.g. teachers!) take refuge in believing the logical (the X ought to) over the statistical (the huh so apparently not X), and thus pass on a bad understanding of statistics to the next generation. To be clear, nothing about this is surprising because statistics is an incredibly young field and in a few hundred years I'm sure we'll all be more statistically literate and comfortable with such reasoning. But we're not there yet. And I don't think this has anything to do with human limitations but just a shift in mentality that needs to take place across a broad fraction of the population to really take hold.
- mjburgess 2y ago
  I'd phrase it as a difference between decision theory and statistics. In almost all cases we're formulating advice, not predictions. In the monty-hall case, we're rarely presented with a person who is going to help us in those circumstances.. so we are motivated to reason to an understanding of the situation in which that help is absent. ie., a better model of the scenario is harder to arrive at, because it involves making ecologically irrational assumptions on behalf of the motivations of others in a context (ie., game show) where suspicion is generally required. It's a perennial annoyance of mine that the weirdest people with the least fit social instincts (relevant academics) go around inveighing on supposed 'ordinary people' prescribing, most often, patterns of thinking that would get everyone killed. What the practice of formulating problems shows is that mathematicians (et al.) are extremely bad at phrasing problems to be intuitive. Indeed, I think they deliberately do the opposite, wishing to 'expose' the idiocy of others and their own superiority. Physicists do not practice physics this way. It would be insane to deliver a lecture where clear physical intuitions can be built, by instead choosing cases where they break down and saying, "ah! so you need the mathematics!". Physicists would, certainly, hold such an activity in contempt. Here we should be formulating these problems to be maximally intuitive, holding their structure constant. Then showing how the formalism helps.

vitus 2y ago

> I'm fundamentally sceptical that animals have, in general, a "poor" understanding of probability. I take the opposite view. Casinos with all of their negative-expected-value (for players) games would not exist, nor would many lotteries, if not for the allure of "oh, but if I win big..." overpowering any rational decision-making. Monty Hall is not challenging as a result of information loss due to translating to natural language. Reading the simulation code is like telling ChatGPT to think about the problem step-by-step: if you picked the door with the car, the host opens one of the goat doors arbitrarily; if you pick a door with a goat, the host opens the other goat door. Further, if you flip the sequence of events, such that the host opens a door containing a goat, and then you pick, you end up with a different probability of success entirely -- 50%, instead of 66.7% (always switch) or 33.3% (never switch)! This played out in the actual game show in real time, where people could observe the actual events directly. Per Monty Hall himself: > He said he was not surprised at the experts' insistence that the probability was 1 out of 2. "That's the same assumption contestants would make on the show after I showed them there was nothing behind one door," he said. "They'd think the odds on their door had now gone up to 1 in 2, so they hated to give up the door no matter how much money I offered." https://en.wikipedia.org/wiki/Monty_Hall_problem#History https://en.wikipedia.org/wiki/Monty_Hall_problem#History

mjburgess 2y ago
I'm not sure you've said anything that disagrees with me. My claim is that the information in the environment that people are sensitive to causes them to fail to accurately model the scenario. The problem they are solving, in their heads, fails to correspond to reality. My claim is: if it did, then they would solve it correctly. This is often phrased, typically by self-important mathematicians, that people's understanding of probability is wrong. This is a very bizarre view, and rather self-aggrandizing as far as probability goes. Probability isn't understanding or formulating the problem. Probability is merely the machinery, very basic, which runs upon a formulation. Do this instead: run one round of a game with 10 doors, and have people observe Monty opening all the relevant doors. Why should it be that if people observe this it becomes obvious, and yet if put in natural language it isn't? Natural language applied to non-intutive ecologies (ie., environments outside those we usually operate in) cause people to fail to model the sitaution correctly. Take casinos: when animals go out foraging they are playing a slot machine. Their addiction to foraging is perfectly ecologically rational. Casions exploit an information asymmetry: people cannot see the long-run house advantage. This has nothing to do with people's "probability" skills, and just their ability to accurately understand the sitatuon. The underlying foolishness behind those who equivocate between "probability" and "modelling scenarios" is the assumption that reality itself has probabilities, and that scenarios are easy to model. It is incredibly difficult to come up with an accurate cognitive model of wtf is going on in a monty hall game. So much so that famous mathematicians, with arbitrarily good probability skills, get the answer wrong. Teaching people "probability" here is beside the point. Animals are acutely good at it when they are presented with an ecologically intutive problem, of arbitary probablistic complexity.
- vitus 2y ago
  Ah. I had focused on the lossiness of natural language much more than the rest of your statement. > Why should it be that if people observe this it becomes obvious, and yet if put in natural language it isn't? It's not clear to me that the general public would come to that conclusion, as opposed to the obvious extension of Hall's observation where contestants thought their odds having gone up upon being shown additional information. To you and me, yes, it might be even more obvious because we're familiar with the core principles. > Take casinos: when animals go out foraging they are playing a slot machine. Their addiction to foraging is perfectly ecologically rational. I'm inclined to disagree here. Surely animals don't just engage on a random walk; they know where they've found berries before, or can smell prey from a distance. And describing it as an addiction feels wrong. Once the animals are full, they don't keep hunting for food at the expense of other life-sustaining activities. > Casinos exploit an information asymmetry: people cannot see the long-run house advantage. Sure. I have also encountered hubris in the form of people who are aware of the house's edge but are confident they can do better than everyone else. There are also betting strategies that have positive expected value (e.g. lose a coin flip? double your bet; win a coin flip? reset to $1) but fail to take into consideration that you have probability = 1 of going bankrupt in the long run, since you don't have infinite money. > The underlying foolishness behind those who equivocate between "probability" and "modelling scenarios" is the assumption that reality itself has probabilities, and that scenarios are easy to model. In many cases, we use probabilities to approximate unknowns. Efficacy of a vaccine? Prevents infection X% of the time, or severe illness Y% of the time. Why not 100%? We don't necessarily know (or maybe it's due to a bunch of complex factors). But we've observed these outcomes for these samples of our population, and reducing it to a single number is "good enough".

kqr 2y ago

> I'm fundamentally sceptical that animals have, in general, a "poor" understanding of probability. I agree with this skepticism, and I think it boils more down to insufficient statistical literacy. Statistical statements (what things are actually like) often stand in contrast to logical reasoning (what things ought to be like) – popularly observed in quips such as "in theory there is no difference between theory and practice." This difference is uncomfortable and so many people (especially those that are better educated and well versed in logical reasoning – e.g. teachers!) take refuge in believing the logical (the X ought to) over the statistical (the huh so apparently not X), and thus pass on a bad understanding of statistics to the next generation. To be clear, nothing about this is surprising because statistics is an incredibly young field and in a few hundred years I'm sure we'll all be more statistically literate and comfortable with such reasoning. But we're not there yet. And I don't think this has anything to do with human limitations but just a shift in mentality that needs to take place across a broad fraction of the population to really take hold.

mjburgess 2y ago
I'd phrase it as a difference between decision theory and statistics. In almost all cases we're formulating advice, not predictions. In the monty-hall case, we're rarely presented with a person who is going to help us in those circumstances.. so we are motivated to reason to an understanding of the situation in which that help is absent. ie., a better model of the scenario is harder to arrive at, because it involves making ecologically irrational assumptions on behalf of the motivations of others in a context (ie., game show) where suspicion is generally required. It's a perennial annoyance of mine that the weirdest people with the least fit social instincts (relevant academics) go around inveighing on supposed 'ordinary people' prescribing, most often, patterns of thinking that would get everyone killed. What the practice of formulating problems shows is that mathematicians (et al.) are extremely bad at phrasing problems to be intuitive. Indeed, I think they deliberately do the opposite, wishing to 'expose' the idiocy of others and their own superiority. Physicists do not practice physics this way. It would be insane to deliver a lecture where clear physical intuitions can be built, by instead choosing cases where they break down and saying, "ah! so you need the mathematics!". Physicists would, certainly, hold such an activity in contempt. Here we should be formulating these problems to be maximally intuitive, holding their structure constant. Then showing how the formalism helps.