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Seeing Theory: A Visual Introduction to Probability and Statistics

seeing-theory.brown.edu

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Re: Seeing Theory: A Visual Introduction to Probability and Statistics

#71

Post by jonsen »

542354234235 wrote: post #68
jonsen wrote: post #60

Earlier quoted context omitted.

> 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.

> 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…

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.

Re: Seeing Theory: A Visual Introduction to Probability and Statistics

#72

Post by jibal »

jonsen wrote: post #71
542354234235 wrote: post #68

Earlier quoted context omitted.

> 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…

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.

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.

Re: Seeing Theory: A Visual Introduction to Probability and Statistics

#73

Post by jibal »

pyhtel wrote: post #63
jibal wrote: post #50

Earlier quoted context omitted.

"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.

"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

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.

Re: Seeing Theory: A Visual Introduction to Probability and Statistics

#74

Post by jonsen »

jibal wrote: post #72
jonsen wrote: post #71

Earlier quoted context omitted.

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.

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.

> 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.

Re: Seeing Theory: A Visual Introduction to Probability and Statistics

#75

Post by ayewo »

roenxi wrote: post #42
Eugeleo wrote: post #31

What textbook(s?) would you recommend for a thorough self-learning of statistics? I’m looking for both intuition _and_ mathematical rigor — not all proofs, but not all fluff either. I’m a bioinformatics student and I will have a semester of combined probability/stats some time this year, but I think that won’t be enough to support me given my preference for DS-based bioinformatics jobs. I’m reading Feller right now f…

Rather than a textbook, I've had success getting a copy of the course notes directly from the stats department. The best textbooks I've read where history of statistics and philosophy of statistics. > I’m reading Feller right now for the probability stuff, but I’m unsure about statistics. Probability is the study of mathematical objects, and nobody is totally sure if any of them exist even in the approximate. Is anyt…

Can you explain this sentence a bit more: "The best textbooks I've read where history of statistics and philosophy of statistics." ?

Are these names of actual books (Google doesn't help) or merely the themes of the stats textbooks you benefited from the most?

Thank you.

Re: Seeing Theory: A Visual Introduction to Probability and Statistics

#76

Post by 542354234235 »

jonsen wrote: post #71
542354234235 wrote: post #68

Earlier quoted context omitted.

> 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…

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.

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).

Re: Seeing Theory: A Visual Introduction to Probability and Statistics

#77

Post by debbiedowner »

st1x7 wrote: post #54
will_pseudonym wrote: post #28

Earlier quoted context omitted.

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?

I see this argument a lot and for some reason it doesn't help me with the intuition at all. If you (wrongly) get caught up on the fact that the remaining door and your pick have the same initial probabilities of being a car, then you'll still think that switching doesn't make a difference even in the million-door case. Here's what works for me: - the switching strategy always gives you the opposite of your initial ch…

When Monty opens doors he uses 2 pieces of information: the door you picked and the correct door.

After he opens 999,998 doors he has given you quite a bit of information. There is a 1/1000000 chance though that he has given you no information (you picked the correct door)

But you're right that thinking about it in partitions also makes sense. You try to pick a partition size 1 that contains the prize, while Monty picks the partition size 999,999, if you agree with his partition and it has the prize you get it

Re: Seeing Theory: A Visual Introduction to Probability and Statistics

#78

Post by mturmon »

jtsuken wrote: post #34
mturmon wrote: post #29

Earlier quoted context omitted.

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…

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 Ha…

> 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.

Re: Seeing Theory: A Visual Introduction to Probability and Statistics

#79

Post by jonsen »

542354234235 wrote: post #76
jonsen wrote: post #71

Earlier quoted context omitted.

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.

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).

> 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.

Re: Seeing Theory: A Visual Introduction to Probability and Statistics

#80

Post by kqr »

jibal wrote: post #73
pyhtel wrote: post #63

Earlier quoted context omitted.

"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

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.

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!

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