Sets and Probability

https://stopa.io/post/243

by stopachka • 6 years ago

96 35 6 years ago

stopa.io

RgueNkeScientst• 6 years ago

This was so satisfying to read.

stopachka3• 6 years ago

Thank you for the kind words :)

canjobear• 6 years ago

The set-based intuitions are a nice guide, but knowing how to turn the crank and churn out the calculations is also crucial for the cases where intuitive thinking doesn't scale.

Tarsul• 6 years ago

yeah, I also liked this guide. Would love more articles about probabilities (but i always loved this topic :)), there's only one exception: the "3 door problem" (Monty Hall problem). In my opinion the general misconception about this problem does not stem from a misunderstanding of probabilites but a misconception about the moderator (who plays by his own rules [as a bad actor] - which is often not really discussed although it changes the whole game). Well, I would read an article about it by this author anyway ;)

Pyramus3• 6 years ago

Not sure what you mean with "bad actor" - the host/moderator is not an adversary, his rules are fixed from the outset (see e.g. Standard Assuptions in [1])

These rules imply that new information is revealed, and based on the new information it is advantageous to switch.

The trickyness of the Monty Hall Problem lies in identifying that indeed new information has been revealed, which seems counter-intuitive.

[1] https://en.wikipedia.org/wiki/Monty_Hall_problem

mannykannot• 6 years ago

In my opinion the general misconception about this problem does not stem from a misunderstanding of probabilites but a misconception about the moderator.

An analysis of the replies to the column in which the puzzle first had wide circulation showed that most of the respondents disputing the correct answer had understood the problem as intended, but still got the probabilities wrong. At that time, the game show on which it was based was well-known, so confusion over the moderator's role (which is not as a "bad actor") was less likely than now.

tmoertel• 6 years ago

Would love more articles about probabilities...

Well, I wrote a few articles on the topic from a Bayesian perspective few years ago:

http://blog.moertel.com/tags/probability.html

(Only one of them is about the Monty Hall problem.)

thehappypm• 6 years ago

Using the counting method for the Monty Hall problem actually works very well. Initially there are 3 possible ways the goats and treasure can be arranged:

goat goat treasure

goat treasure goat

treasure goat goat

You pick a door -- let's say it's the first door. Monty opens one of the other 2 to reveal a goat, leaving one other door. This modifies the possible outcomes:

goat treasure

treasure goat

Using the counting approach, there is 1 outcome where door 1 has the treasure, and 2 outcomes where the other door has the treasure. 2/3 chance of finding the treasure if you switch.

jhncls• 6 years ago

It's indeed a nice guide. One place where the intuition stops unexpectedly, is calculating the probability of getting 2 reds balls out (without replacement). In the situation with all balls together, the chance would be 1/4. But with two separate boxes, the probability would change to 4999/20000 (1/2 * (1/100 * (0 + 1/2 * 99/100) + 99/100 * (1/2 * 1/100 + 1/2 * 98/99))).

cleansingfire• 6 years ago

There's no 1/4 here. (1/2) (99/199) = 99/398 First ball removed changes the odds for second draw.

BeetleB2• 6 years ago

It's really not just a guide. It is practically the basis. Yes, knowing how to do the calculations will make life much simpler, but I often find the usual calculation rules are easy to misuse. I often derive things from scratch using the set based axioms.

vertere3• 6 years ago

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.

wizzwizz4• 6 years ago

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

vertere3• 6 years ago

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.

petters• 6 years ago

Ironically, this is the sort of probability theory Nassim Taleb hates. :-)

atdixon• 6 years ago

Exactly. This is the brand of neutered, ludic probability on which The Black Swan is a holistic dump. Very ironic indeed that it leads with Taleb as the inspiration.

sAbakumoff• 6 years ago

References to textbooks that OP read would be a nice addition to the essay

stopachka3• 6 years ago

My favorite was this textbook:

https://www.amazon.com/gp/product/1292025042/ref=ox_sc_act_t...

This one was more dense, but very good in the beginning if you take the time: https://www.amazon.com/gp/product/0070484686/ref=ppx_od_dt_b...

---

Will update post later, but wanted to share with you now : } -- thanks for taking the time to read!

yboris• 6 years ago

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" approach. A great illustration is one about a 99% accurate medical test that claims a patient is positive. The probability the patient has the disease is not 99% (over 60% of Harvard doctors get this question wrong). The authors show how reframing the question with a sample population (10,000 people for example) and the relative prevalence of the disease (and noticing the false negatives) the problem becomes almost trivial to calculate.

[0] https://www.amazon.com/Epistemology-Psychology-Judgment-Mich...

fredgrott• 6 years ago

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

Sharlin• 6 years ago

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)

benrbray2• 6 years ago

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 measure spaces) onwards, the book is full of great insight.

A good one on probability & stochastic processes is "Measure Theory and Probability Theory" by Athreya & Lahiri.

[1] https://terrytao.files.wordpress.com/2011/01/measure-book1.p... [2] https://www.springer.com/gp/book/9780387329031

t_serpico• 6 years ago

How useful do you think measure theory is from a practical perspective? I've seemed to get by without more or less during my career (which is pretty stats heavy).

jordan_curve• 6 years ago

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.

BeetleB2• 6 years ago

I don't recall how I learned it in high school, but for my undergrad statistics & probability course, sets were very much the basis for everything. I'm surprised that this isn't the norm ...?

zokier• 6 years ago

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?

senthil_rajasek• 6 years ago

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?

somethingsome• 6 years ago

(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.

stopachka3• 6 years ago

Will do, thank you! : }

Sets and Probability

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