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"Don't trust your intuition". This should be the basis for all teaching in statistics and probability. If all goes wrong, it should be the one thing everyone re

"Don't trust your intuition". This should be the basis for all teaching in statistics and probability. If all goes wrong, it should be the one thing everyone remembers from their statistics education. And yet year after year, everyone is starting with E(X)=sum(x*P(x)) and has no idea what it was about afterwards. With calculus and linear algebra your gut feel is about right no average. You can quickly get a feel for trajectories, acceleration and distances (derivatives and integrals), areas, volumes, amounts, etc. But on probability your gut-feel will always fool you. In the end, you see a handful of math bloggers bemoaning the lack of education in probability and the nonsense being discussed by journalists and politicians. And it hardly matters whether it's an election or a pandemic. The lack of understanding of uncertainty and the false belief that one can reason about these without looking at the numbers too closely is dangerous. Sorry about the rant. But... Dear creator of seeing-theory.brown.edu, if there is one thing you could change about the project to make it different and infinitely more useful: Please start the first chapter with the goat problem[1], then go through a couple of examples from chapter 10 in Thinking Fast and Slow[2], the discuss information (maybe with a simplified version of Mendel's pea experiment[3]), discuss distributions and leave expectations and variances for much-much later. [1]: https://en.wikipedia.org/wiki/Monty_Hall_problem https://en.wikipedia.org/wiki/Monty_Hall_problem [2]: https://en.wikipedia.org/wiki/Thinking,_Fast_and_Slow https://en.wikipedia.org/wiki/Thinking,_Fast_and_Slow [3]: https://www.sciencelearn.org.nz/resources/1999-mendel-s-experiments https://www.sciencelearn.org.nz/resources/1999-mendel-s-expe...

whatever1 6y ago
We are very good at finding correlations. It is still very hard to prove causality in natural phenomena from experiments, specially when we cannot control them. This became blatantly obvious in the covid outbreak where nobody had a clue for months about whether masks would help or not. Edit to clarify: It is very hard to prove to causality and be sure that you did not mess up.
- jtsuken 6y ago
  You are right about people confusing causality and correlation. Otherwise this site wouldn't be so funny: https://www.tylervigen.com/spurious-correlations https://www.tylervigen.com/spurious-correlations You are wrong about people being good at finding correlations. I rarely met people who can process a sufficiently large sample size in their memory to calculate any significant correlation results. Whereas guessing correlations from charts exposes you to a number of optical illusions, which will fool the brain into seeing things that don't exist. There may be a propensity to make more type 2 errors and see correlations between any random things such as 5G and COVID, but I haven't seen any research on that.
wodenokoto 6y ago
Why are you calling the Monty Hall problem for the goat problem? It is known in academic circles as Monty hall and when it pops up in popular media, it is also referred to as Monty hall.
- will_pseudonym 6y ago
  From Wikipedia[0] ... It became famous as a question from a reader's letter quoted in Marilyn vos Savant's "Ask Marilyn" column in Parade magazine in 1990 (vos Savant 1990a): 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? [0] https://en.wikipedia.org/wiki/Monty_Hall_problem https://en.wikipedia.org/wiki/Monty_Hall_problem
- jtsuken 6y ago
  Thanks for pointing this out. While Google and Wikipedia confidently redirect me to the Monty Hall article, which does mention a goat at some point, the common name for it in English is "Monty Hall Problem". In other languages it's a three-door-problem or the goat problem, but given that it's such a good example for so many things in psychology and maths, I should be using the most common name in each language.
will_pseudonym 6y ago
On the topic of the Monty Hall problem, what helped me "believe" it more was if you change it to 1,000,000 doors, still with only 1 car, and the rest goats. You choose 1 door. The host then opens up 999,998 other doors, which all contain goats. So there are 2 doors left. Your door, and the only other door the host didn't open. Do you feel at a gut level that you should switch?
- em500 6y ago
  Many people have suggested this "intuitive" explanation. But it's not at all clear or intuitive that jumping from 3 to 1,000,000 doors should lead the host to open 999,998 other doors rather than 1 other door.
  - jibal 6y ago
    "But it's not at all clear or intuitive that jumping from 3 to 1,000,000 doors should lead the host to open 999,998 other doors rather than 1 other door." It SHOULD be clear, because you have two givens: 1) Monty never reveals the car. 2) He opens all the doors except 1.
    - pyhtel 6y ago
      "2) He opens all the doors except 1" How is this a given exactly? In the original problem he only opens 1 other door. Now that also happens to be all doors except 1, but from just the 3 door problem that seems more coincidental than a fundamental part to the question
      - jibal 6y ago
        It's a given by the person who mentioned 999,998 doors. I think you're missing the point, but I won't pursue this further.
        kqr 6y ago
        Sorry, but you are the one missing the point. The person who mention 999,998 doors didn't give any reasoning for why that would be the logical extension of the problem. Obviously, you and I know it is, but the person grappling with the Monty Hall problem is right in not being convinced of that just because someone says it is!
mturmon 6y ago
Wow, strong disagree. Once you develop intuition, probability is really quite intuitive. This kind of course should be working to develop this intuition — like the conditional probability examples and the CLT examples. The computational examples inline really help here. The Monte Hall problem is more of a curiosity than a fundamental principle! (Was a TA in undergrad engineering probability for 2 years, saw my share of learners.)
- jtsuken 6y ago
  I can't argue with "once you develop intuition, probability is intuitive". I was arguing that lessons starting with E(X)=... basically stop the majority of people from getting to the point, where they see how their "initial intuition" is wrong. Convincing as many people as possible that statistical intuition is not something we are born with should be the key priority of any probability and statistics class. Monte Hall was one example. The birthday problem and the base rate fallacy are two more [1][2]. The result seems obvious but most people get these wrong. With a couple of papers or books by Kahneman and Tversky in hand we can generate an almost infinite list of simple statistics/probability questions, which most people get wrong. Let people make some mistakes, before dumping the theory on them. [1]https://en.wikipedia.org/wiki/Base_rate_fallacy https://en.wikipedia.org/wiki/Base_rate_fallacy [2]https://en.wikipedia.org/wiki/Birthday_problem https://en.wikipedia.org/wiki/Birthday_problem
  - jonsen 6y ago
    > Monte Hall was one example. Monty Hall is not a good example, unless it is explicitly stated that Monty knows where the car is and that he deliberately opens a door with a goat. Just look at the discussions in the comment here.
  - mturmon 6y ago
    > Convincing as many people as possible that statistical intuition is not something we are born with should be the key priority of any probability and statistics class. Again, strong disagree. Probability has been understood at a quantitative level since Laplace (1812). Modern measure-theoretic probability dates from Kolmogorov's foundational work (1933). All these years later, we really know this stuff. Specifically: A lot of general-purpose, powerful tools have been developed. Distribution theory, the strong law of large numbers, the CLT, maximum likelihood, L2 theory for estimation. Depending on your goals, these or related tools are capable of addressing a wide range of problems. The priority of the first few courses should be to impart mastery of a selection of these general-purpose tools, so that students know how to analyze problems probabilistically. This is where intuition comes from. Gotcha problems like Monte Hall are not getting you to this goal! One could argue that MHP can motivate the notion of conditioning, but I think fundamentally the MHP is verbal legerdemain. That is, you state the problem such that the conditioning is implicit in the actions, and people don't notice it. Recall that the questioner obtains "victory" when, after presenting the problem, the answerer is confused and gives the wrong answer. I don't like that approach as a teaching tool. I'm also skeptical of the Birthday Problem and the Kahneman-Tversky surprises. I see value in these surprising conundrums (the Birthday Problem is in volume 1 of Feller, so it has a pedigree) only to the extent that they motivate the utility of general-purpose analytic tools. They are an appetizer, not the main dish.
WildParser 6y ago
My professor for statistics (he was quite famous in the field) talked about Monty Hall, but made clear that he will not give a solution because of science-political reasons.
- will_pseudonym 6y ago
  I am incredibly curious what he meant by "science-political reasons."
  - WildParser 6y ago
    Me too, but he refused to explain. I think there must have been a time where choosing a side was able to end friendships.
  - hinkley 6y ago
    Bringing up Monty Hall at a table full of tech people has always resulted in an argument that will not end until one person gets the rest of us to admit we are wrong and changing doors is the same probability as staying with the same door.
mtzet 6y ago
The mathematics behind probability and statistics is about as ripe for intuition as calculus and linear algebra. A lot of it really comes down to counting in probability (calculus/measure theory for the continuous case) and quantifying properties about probability distributions for statistics. The really hard part is the modelling part, where you transform the problem to a mathematical statement and vice versa. It's very easy to misinterpret both the problem in terms of mathematics and the mathematical result in terms of the problem. All the wrong answers to brain teasers like the monty hall problem, the tuesday boy problem etc., are right answers to the wrong question. Unfortunately, in education we do not seem to want to discuss the modelling part on equal terms with the theory. We seem to be okay with solving the entire problem, or solving just the theoretical part with no regards to the application, but expressing just the mathematical problem to be solved is never appreciated. In a calculus setting, this could be deriving the answer to some physical problem depends on the solution of some partial differential equation -- even if you do not have the tools to solve it outright. My guess is that it's just easier to teach theory with clear cut answers. Modelling the real world is ambiguous and hard.
- swiley 6y ago
  It's because math courses are crazy expensive and teaching both modelling and theory together would mean taking way longer (read costing way more) or making the failure rate (cost) way higher.
mnky9800n 6y ago
I'm always wondering the answer to this question: Why does multiplication coupled with some sort of integral calculus work the way it does? We multiply to get moments of a distribution, we multiply to convolve, we multiply to get the work done on an object. I suppose the answer is multiplication allows us to scale some function f(x) with some function g(x). But I guess I want something deeper and I feel like I'm missing it.
1e-9 6y ago
> But on probability your gut-feel will always fool you. Yes, this is especially true when first learning; however, one can still develop intuition so that it serves as an invaluable motivator and guide through difficult problems.
unwaryquerier 6y ago
"The law of Small Numbers" chapter 10?

whatever1 6y ago

We are very good at finding correlations. It is still very hard to prove causality in natural phenomena from experiments, specially when we cannot control them. This became blatantly obvious in the covid outbreak where nobody had a clue for months about whether masks would help or not. Edit to clarify: It is very hard to prove to causality and be sure that you did not mess up.

jtsuken 6y ago
You are right about people confusing causality and correlation. Otherwise this site wouldn't be so funny: https://www.tylervigen.com/spurious-correlations https://www.tylervigen.com/spurious-correlations You are wrong about people being good at finding correlations. I rarely met people who can process a sufficiently large sample size in their memory to calculate any significant correlation results. Whereas guessing correlations from charts exposes you to a number of optical illusions, which will fool the brain into seeing things that don't exist. There may be a propensity to make more type 2 errors and see correlations between any random things such as 5G and COVID, but I haven't seen any research on that.

wodenokoto 6y ago

Why are you calling the Monty Hall problem for the goat problem? It is known in academic circles as Monty hall and when it pops up in popular media, it is also referred to as Monty hall.

will_pseudonym 6y ago
From Wikipedia[0] ... It became famous as a question from a reader's letter quoted in Marilyn vos Savant's "Ask Marilyn" column in Parade magazine in 1990 (vos Savant 1990a): 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? [0] https://en.wikipedia.org/wiki/Monty_Hall_problem https://en.wikipedia.org/wiki/Monty_Hall_problem
jtsuken 6y ago
Thanks for pointing this out. While Google and Wikipedia confidently redirect me to the Monty Hall article, which does mention a goat at some point, the common name for it in English is "Monty Hall Problem". In other languages it's a three-door-problem or the goat problem, but given that it's such a good example for so many things in psychology and maths, I should be using the most common name in each language.

will_pseudonym 6y ago

On the topic of the Monty Hall problem, what helped me "believe" it more was if you change it to 1,000,000 doors, still with only 1 car, and the rest goats. You choose 1 door. The host then opens up 999,998 other doors, which all contain goats. So there are 2 doors left. Your door, and the only other door the host didn't open. Do you feel at a gut level that you should switch?

em500 6y ago
Many people have suggested this "intuitive" explanation. But it's not at all clear or intuitive that jumping from 3 to 1,000,000 doors should lead the host to open 999,998 other doors rather than 1 other door.
- jibal 6y ago
  "But it's not at all clear or intuitive that jumping from 3 to 1,000,000 doors should lead the host to open 999,998 other doors rather than 1 other door." It SHOULD be clear, because you have two givens: 1) Monty never reveals the car. 2) He opens all the doors except 1.
  - pyhtel 6y ago
    "2) He opens all the doors except 1" How is this a given exactly? In the original problem he only opens 1 other door. Now that also happens to be all doors except 1, but from just the 3 door problem that seems more coincidental than a fundamental part to the question
    - jibal 6y ago
      It's a given by the person who mentioned 999,998 doors. I think you're missing the point, but I won't pursue this further.
      - kqr 6y ago
        Sorry, but you are the one missing the point. The person who mention 999,998 doors didn't give any reasoning for why that would be the logical extension of the problem. Obviously, you and I know it is, but the person grappling with the Monty Hall problem is right in not being convinced of that just because someone says it is!

mturmon 6y ago

Wow, strong disagree. Once you develop intuition, probability is really quite intuitive. This kind of course should be working to develop this intuition — like the conditional probability examples and the CLT examples. The computational examples inline really help here. The Monte Hall problem is more of a curiosity than a fundamental principle! (Was a TA in undergrad engineering probability for 2 years, saw my share of learners.)

jtsuken 6y ago
I can't argue with "once you develop intuition, probability is intuitive". I was arguing that lessons starting with E(X)=... basically stop the majority of people from getting to the point, where they see how their "initial intuition" is wrong. Convincing as many people as possible that statistical intuition is not something we are born with should be the key priority of any probability and statistics class. Monte Hall was one example. The birthday problem and the base rate fallacy are two more [1][2]. The result seems obvious but most people get these wrong. With a couple of papers or books by Kahneman and Tversky in hand we can generate an almost infinite list of simple statistics/probability questions, which most people get wrong. Let people make some mistakes, before dumping the theory on them. [1]https://en.wikipedia.org/wiki/Base_rate_fallacy https://en.wikipedia.org/wiki/Base_rate_fallacy [2]https://en.wikipedia.org/wiki/Birthday_problem https://en.wikipedia.org/wiki/Birthday_problem
- jonsen 6y ago
  > Monte Hall was one example. Monty Hall is not a good example, unless it is explicitly stated that Monty knows where the car is and that he deliberately opens a door with a goat. Just look at the discussions in the comment here.
- mturmon 6y ago
  > Convincing as many people as possible that statistical intuition is not something we are born with should be the key priority of any probability and statistics class. Again, strong disagree. Probability has been understood at a quantitative level since Laplace (1812). Modern measure-theoretic probability dates from Kolmogorov's foundational work (1933). All these years later, we really know this stuff. Specifically: A lot of general-purpose, powerful tools have been developed. Distribution theory, the strong law of large numbers, the CLT, maximum likelihood, L2 theory for estimation. Depending on your goals, these or related tools are capable of addressing a wide range of problems. The priority of the first few courses should be to impart mastery of a selection of these general-purpose tools, so that students know how to analyze problems probabilistically. This is where intuition comes from. Gotcha problems like Monte Hall are not getting you to this goal! One could argue that MHP can motivate the notion of conditioning, but I think fundamentally the MHP is verbal legerdemain. That is, you state the problem such that the conditioning is implicit in the actions, and people don't notice it. Recall that the questioner obtains "victory" when, after presenting the problem, the answerer is confused and gives the wrong answer. I don't like that approach as a teaching tool. I'm also skeptical of the Birthday Problem and the Kahneman-Tversky surprises. I see value in these surprising conundrums (the Birthday Problem is in volume 1 of Feller, so it has a pedigree) only to the extent that they motivate the utility of general-purpose analytic tools. They are an appetizer, not the main dish.

WildParser 6y ago

My professor for statistics (he was quite famous in the field) talked about Monty Hall, but made clear that he will not give a solution because of science-political reasons.

will_pseudonym 6y ago
I am incredibly curious what he meant by "science-political reasons."
- WildParser 6y ago
  Me too, but he refused to explain. I think there must have been a time where choosing a side was able to end friendships.
- hinkley 6y ago
  Bringing up Monty Hall at a table full of tech people has always resulted in an argument that will not end until one person gets the rest of us to admit we are wrong and changing doors is the same probability as staying with the same door.

mtzet 6y ago

The mathematics behind probability and statistics is about as ripe for intuition as calculus and linear algebra. A lot of it really comes down to counting in probability (calculus/measure theory for the continuous case) and quantifying properties about probability distributions for statistics. The really hard part is the modelling part, where you transform the problem to a mathematical statement and vice versa. It's very easy to misinterpret both the problem in terms of mathematics and the mathematical result in terms of the problem. All the wrong answers to brain teasers like the monty hall problem, the tuesday boy problem etc., are right answers to the wrong question. Unfortunately, in education we do not seem to want to discuss the modelling part on equal terms with the theory. We seem to be okay with solving the entire problem, or solving just the theoretical part with no regards to the application, but expressing just the mathematical problem to be solved is never appreciated. In a calculus setting, this could be deriving the answer to some physical problem depends on the solution of some partial differential equation -- even if you do not have the tools to solve it outright. My guess is that it's just easier to teach theory with clear cut answers. Modelling the real world is ambiguous and hard.

swiley 6y ago
It's because math courses are crazy expensive and teaching both modelling and theory together would mean taking way longer (read costing way more) or making the failure rate (cost) way higher.

mnky9800n 6y ago

I'm always wondering the answer to this question: Why does multiplication coupled with some sort of integral calculus work the way it does? We multiply to get moments of a distribution, we multiply to convolve, we multiply to get the work done on an object. I suppose the answer is multiplication allows us to scale some function f(x) with some function g(x). But I guess I want something deeper and I feel like I'm missing it.

1e-9 6y ago

> But on probability your gut-feel will always fool you. Yes, this is especially true when first learning; however, one can still develop intuition so that it serves as an invaluable motivator and guide through difficult problems.

unwaryquerier 6y ago

"The law of Small Numbers" chapter 10?