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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

#81

Post by mturmon »

jibal wrote: post #51
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…

"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 conditiona…

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.

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

#82

Post by 1e-9 »

jtsuken wrote: post #14

"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…

> 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.

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

#83

Post by hinkley »

will_pseudonym wrote: post #37
WildParser wrote: post #30

Earlier quoted context omitted.

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.

I am incredibly curious what he meant by "science-political reasons."

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.

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

#84

Post by TurtleDove03 »

will_pseudonym wrote: post #28
jtsuken wrote: post #14

"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…

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?

Rather than going through this convoluted kinda similar problem, I find it easier to stick to the original one.

Get a piece of paper. Draw all possible outcomes, 9 total. ( Car is behind door 1 you pick 1, Car is behind door 1 you pick 2...). 3 of the 9 result in success.

Now draw the outcomes again but switch every time. 6 out of 9 outcomes are a success.

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

#85

Post by will_pseudonym »

kqr wrote: post #80
jibal wrote: post #73

Earlier quoted context omitted.

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!

The rationale for opening 999,998 doors in my example versus the 1 door left, is that in both examples, Monty Hall opens every door except the one you're on, and 1 other door. It happens that in the normal Monty Hall, if you open every door except the one you're on, and 1 other door, you have only opened one door.

Monty Hall is asking you a simple question, whether or not you should switch, and so in my example of 1,000,000 whether or not you open 999,998 doors, or 1 door, you will always have worse odds to win if you don't switch to another door. Removing 999,998 doors just takes the proposition to an extreme.

Another component to utilize one's intuition using the 999,998 example, would be to imagine the game being played 3 times in a row. What are the odds that not switching will help you? So basically, not switching is disregarding everything Monty Hall is doing. You are either behind a door or you are not. You don't switch. If that is how you play the game, your chance of choosing right when not switching is 1/1,000,000 each game, or 1/10^18 for it to happen 3 times in a row. Now, consider what Monty is doing. He's removing every chair but two, yours and another. If the odds of you winning are 1/1,000,000 if you don't switch, What are the odds of doing _the opposite_? Since there are only two options, the probability of winning if you switch is 1-1/1,000,000, or 999,999/1,000,000, as the sum of the probabilities of all possible events has to add up to 1.

The "999,998" chairs removed example is an attempt at making the dichotomy between "stay" and "switch" more extreme, so that you would feel it in your gut rather than trying to mentally account for the moving pieces.

I'm always interested in improving my ability to explain these kinds of phenomena, and I appreciate the pointing out of why the dots don't get connected for some with the example.

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

#86

Post by will_pseudonym »

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

pyhtel, I gave it a go at explaining the rationale in this comment I made below: https://news.ycombinator.com/item?id=24643272

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

#87

Post by will_pseudonym »

em500 wrote: post #47
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?

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.

em500, I gave it a go at explaining the rationale in this comment I made below: https://news.ycombinator.com/item?id=24643272

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

#88

Post by jmalicki »

ayewo wrote: post #75
roenxi wrote: post #42

Earlier quoted context omitted.

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.

Not the above poster, but I concur, and recommend Jaynes' "Probability: the logic of science" for the philosophy and history, and "Breakthroughs in statistics" volumes 1 and 2 for the history as told through original foundational papers, from the 1700s on.

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

#89

Post by jtsuken »

mturmon wrote: post #78
jtsuken wrote: post #34

Earlier quoted context omitted.

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…

> 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.

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

#90

Post by em500 »

will_pseudonym wrote: post #85
kqr wrote: post #80

Earlier quoted context omitted.

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!

The rationale for opening 999,998 doors in my example versus the 1 door left, is that in both examples, Monty Hall opens every door except the one you're on, and 1 other door. It happens that in the normal Monty Hall, if you open every door except the one you're on, and 1 other door, you have only opened one door. Monty Hall is asking you a simple question, whether or not you should switch, and so in my example of 1,…

> The rationale for opening 999,998 doors in my example versus the 1 door left, is that in both examples, Monty Hall opens every door except the one you're on, and 1 other door. It happens that in the normal Monty Hall, if you open every door except the one you're on, and 1 other door, you have only opened one door.

And the rationale for opening 1 other door in the million door example is that in both examples the host is opening 1 other door. The normal Monty Hall problem is usually formulated such that the host opens 1 other door, not that he opens all other doors. As you noted, the two formulations are equivalent in with 3 doors, but with more than 3 doors, they're not. I just don't see why it's "intuitive" that if the number of doors is increased, the natural extension of the game is that the host opens all other doors that don't have the prize. In fact I'd argue the opposite.

Imagine an actual Monty Hall game with 4 doors. The contestant opens 1 door with a goat, and the host might open (a) 1 other door with a goat or (b) 2 other doors with a goat. Both are valid, reasonable, but different extensions of the game. In both versions, the best strategy for the contestant is to switch[1], because in both versions the host is giving her extra information. But in version (b) he's giving her much more information than in version (a). Of course it's much easier to intuit in version (b) that switching is better, but it's not clear to me why version (b) rather than (a) is the natural 4-door analog to the 3-door Monte Hall game. If a 4 door version were played in real life, it's far more likely IMO that version (a) would be played. In (a) the host gives a little bit of extra info where the prize is, without giving away the solution. And in this case you need a much better model to see why this is the case in stead of relying on intuition and analogy.

[1] A usually unstated assumption in most formulations is that the host must open another door with a goat if the contestant initially chooses a wrong door. In an actual TV show the host will likely have discretion whether he opens another door at all, to increase suspension and not become predictable in repeated games. In this case the problem becomes much more difficult as you need to model the strategy of the host. All of thise and pretty much any other solutions and explanations on pretty much every forum is already extensively documented in Wikipedia (https://en.wikipedia.org/wiki/Monty_Hall_problem#Other_host_...).

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