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.
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.
> 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.
> Probability has been understood at a quantitative level since Laplace [...] and Kolmogorov.
Which indicates it is roughly as hard as partial differential equations, the theory of relativity and just a tiny bit easier than some of the quantum mechanics.
This is pretty unintuitive for a subject, which mostly relies on multiplication and addition.
The dozen or so posts discussing the intuition of the Monty Hall Problem are a case in point.
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 [2]https://en.wikipedia.org/wiki/Birthday_problem