[mturmon]

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]

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 [2]https://en.wikipedia.org/wiki/Birthday_problem

[jonsen]

> 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]

> 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.

[jtsuken]

> 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.

> They are an appetizer, not the main dish.

This is certainly true.

[jibal]

"Once you develop intuition, probability is really quite intuitive"

That's a tautology.

Plenty of studies, such as the work by Kahneman and Tversky, show that humans by default have incorrect statistical intuitions. These faulty intuitions are hard to overcome, even by a considerable amount of training.

> The Monte Hall problem is more of a curiosity than a fundamental principle!

It's quite straightforward conditional probability. That so many people, including trained mathematicians, get it wrong is quite illustrative. And it's not unique ... the coins and drawers problem is similar, and one can craft many others. MH is not a mere curiosity, it's simply well known.

[jonsen]

> It's quite straightforward ...

No it is not, 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.

[542354234235]

> unless it is explicitly stated that Monty knows where the car is and that he deliberately opens a door with a goat.

That has been part of the explicit problem ever since it was first presented back in 1975.

"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?"

[jonsen]

It does not clearly say that it is Monty’s procedure to

- allways open a door

- allways open a door with a goat

- and not open a door at random

Often discussion of the solution reveals that this is not clear.

[542354234235]

There are three doors. You have picked one, leaving two other doors. It absolutely explicitly says he opens one door. The only options are a goat or a car. If it was a car, you would have lost already and so there is no problem. If it was random, you still get the same information (what is behind one of the unpicked doors).

[jonsen]

> If it was random, you still get the same information

No, if both you and Monty pick a door at random, there’s 1/3 chance of a car behind each door. If Monty’s door reveals a goat, it’s 50/50 for the remaining two doors. It’s mandatory to specify Monty’s procedure precisely.

[jibal]

This is goalpost moving that has nothing to do with the original point. If people misunderstand the conditions of the problem, that has nothing to do with intuitions about probability.

I won't respond further.

[jonsen]

> nothing to do with the original point.

Agree, but my point is that the Monty puzzle is a bad example to use educational if not careful.

[mturmon]

It is a tautology, but we are studying teaching, so maybe that's not unexpected? ;-)

My point is that the goal of the course should be to understand principles, not to teach people that their existing intuition is faulty. Who cares about their prior condition of ignorance?

For more, see my reply nearby.