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Comment by canjobear | Hacker News Reader

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.

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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.

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))).

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

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 ;)

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

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.

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

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_...

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.

> 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.

> 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.)

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