Ironically, this is the sort of probability theory Nassim Taleb hates. :-)
Sets and Probability
11–20 of 38 posts
Re: Sets and Probability
#12Earlier quoted context omitted.
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:…
Good on wikipedia for having them, but in my experience those "standard assumptions" are often left out when stating the problem.
I think the assumption that tricks most people is that the host will never eliminate the door with the prize. Since this makes sense for a real TV show (it would kind of crappy from the host to eliminate the door with the prize) sometimes it goes unmentioned.
But, even if you highlight the assumption, for most people it is not straightforward how to use this information.
Re: Sets and Probability
#13The 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.
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 a…
> 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.
Re: Sets and Probability
#14References to textbooks that OP read would be a nice addition to the essay
Re: Sets and Probability
#15The 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.
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 a…
> 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.)
Re: Sets and Probability
#16Earlier quoted context omitted.
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:…
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).
Re: Sets and Probability
#17[deleted]
Re: Sets and Probability
#18One 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...
Re: Sets and Probability
#19The 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.
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 a…
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
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.
Re: Sets and Probability
#20Probability 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].