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Sets and Probability

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Re: Sets and Probability

#31

Post by jordan_curve »

benrbray wrote: post #21
Sharlin wrote: post #20

Probability theory with sets actually generalizes well to infinite and even uncountable cardinalities—rather than counting events you just have to switch to the more general notion of the measure of a (sub)set [1]. [1] https://en.wikipedia.org/wiki/Measure_(mathematics)

For anyone looking properly (i.e. with proofs) learn measure theory, I highly recommend "Introduction to Measure Theory" [1] by Terrence Tao (has an excellent collection of exercises -- they're crucial to the reading!) I think it's OK to skim the first few chapters about constructing Lebesgue measure from Jordan measure, as it makes the whole topic appear more difficult than it needs to be. From Chapter 4 (abstract m…

I’d just make a slight correction in case anyone was confused. The book actually only has two chapters, one on the Lebesgue Measure and one on Abstract Measures. When the person above me said skim the first few chapters they meant sections of the first chapter. I’ll admit it’s a somewhat confusing way to lay out the book.

I also can’t recommend this text enough for anyone looking to learn measure theory. Tao is not only one of the brightest research mathematicians but also is extremely good at communicating and framing concepts.

Re: Sets and Probability

#32

Post by senthil_rajasek »

zokier wrote: post #26

I'm bit confused, isn't counting desirable outcomes vs possible outcomes standard way of teaching probability? How is nCr/nPr stuff explained if not in the context of counting outcomes? I'm also confused where does set theory come in here?

To begin, Sample Space of Events is a Set.

Simple cases like probability of A or A's complement etc., all rely on basic set theory.

The rabbit hole is sample spaces that are really really large.

What happens to probability distribution of desired events in those cases?

Re: Sets and Probability

#33

Post by fredgrott »

yboris wrote: post #18

One of my favorite books Epistemology and Psychology of Human Judgment by Bishop & Trout [0] argues that humans are bad at making judgments, and what we want is reliable methods for arriving at truth. The authors of the book provide several heuristics that yield better outcomes than typical strategies people use. With respect to probabilities, since humans, on average, suck at it, they recommend a "frequentist" appro…

note, folks the book can also be found on archive.org as well

Re: Sets and Probability

#34

Post by vertere »

wizzwizz4 wrote: post #8
vertere wrote: post #6

This method only works because the boxes have the same number of balls. To calculate probabilities by counting outcomes you have to start with outcomes that all have equal probabilities.

It's a special-case of a general principle that works regardless of the weighting. Instead of integer counting, do real counting with unity.

Sure you can generalize it by weighting the outcomes by their probabilities (not mentioned in the article). But how are you going to calculate those probabilities any more easily than you can solve the posed problem?

The more "intuitive" way of generalizing it (integer counting) does not work in general. I think the example is likely to be misleading to anyone who doesn't already understand this stuff.

As suggested by the footnote, the reason the given example can be solved elegantly is because of the symmetry.

Re: Sets and Probability

#35

Post by vertere »

Pyramus wrote: post #16
im3w1l wrote: post #10

Earlier quoted context omitted.

Good on wikipedia for having them, but in my experience those "standard assumptions" are often left out when stating the problem.

Not really in my experience, I've never come across the problem without being stated explicitly how the moderator acts - it would be a different problem. What I think is true is that people underestimate the role of the host (i.e. the rules according to which he behaves to).

The first statement of the problem in the wikipedia page (from "Ask Marilyn") does not state it explicitly (though it does say the host knows what's behind the doors). There is a reason wikipedia refers to them as "assumptions".

Or from [0]: "The standard annunciation of the MH problem, does not make explicit what I am assuming here: namely, that Monty will always open a door which does not contain a prize."

[0]: https://www.montyhallproblem.com/#F2

Re: Sets and Probability

#36

Post by Pyramus »

vertere wrote: post #35
Pyramus wrote: post #16

Earlier quoted context omitted.

Not really in my experience, I've never come across the problem without being stated explicitly how the moderator acts - it would be a different problem. What I think is true is that people underestimate the role of the host (i.e. the rules according to which he behaves to).

The first statement of the problem in the wikipedia page (from "Ask Marilyn") does not state it explicitly (though it does say the host knows what's behind the doors). There is a reason wikipedia refers to them as "assumptions". Or from [0]: "The standard annunciation of the MH problem, does not make explicit what I am assuming here: namely, that Monty will always open a door which does not contain a prize." [0]: htt…

While you are technically correct, it's implicitly stated ("game show") and correctly understood by the majority of readers. Take it from Marilyn vos Savant herself:

"Virtually all of my critics understood the intended scenario. I personally read nearly three thousand letters (out of the many additional thousands that arrived) and found nearly every one insisting simply that because two options remained (or an equivalent error), the chances were even. Very few raised questions about ambiguity, and the letters actually published in the column were not among those few." [0]

The crux of the problem is the counter-intuitive nature of probabilities and information, possible choices and choices that could have been. It's a difficult problem, purported to have tripped up even Paul Erdős.

[0] https://en.wikipedia.org/wiki/Monty_Hall_problem#Other_host_...

Re: Sets and Probability

#37

Post by somethingsome »

(From the name of OP I deduce he may read this.)

You should read the first and second chapter of 'Probability Theory: the Logic of Science' by Jaynes. It offers a derivation of probability theory using only Logic and Boolean algebra as foundations. I found it delightful, I hope you will also enjoy it :)

It's somewhat more intensive than Taleb, but the derivations are really beautiful.

Re: Sets and Probability

#38

Post by stopachka »

somethingsome wrote: post #37

(From the name of OP I deduce he may read this.) You should read the first and second chapter of 'Probability Theory: the Logic of Science' by Jaynes. It offers a derivation of probability theory using only Logic and Boolean algebra as foundations. I found it delightful, I hope you will also enjoy it :) It's somewhat more intensive than Taleb, but the derivations are really beautiful.

Will do, thank you! : }
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