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Douglas_Knight comments on Bayes' Theorem Illustrated (My Way) - Less Wrong

126 Post author: komponisto 03 June 2010 04:40AM

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Comment author: Douglas_Knight 09 June 2010 03:53:54AM *  0 points [-]

Robin's description is correct. I'm not sure what you're saying.

ETA: this thread has gotten ridiculous. I'm deleting the rest of my comments on it. The best source for info on Monty Hall is youtube. He does everything. One thing that makes it rather different is that it is usually not clear how many good and bad prizes there are.

Comment author: Jack 09 June 2010 12:09:57PM 4 points [-]

I'm really shocked by the reactions of the mathematicians. I remember solving that problem in like the third week of my Intro to Computer Science Class. And before that I had heard of it and thought through why it was worth switching. I didn't realize it caused so much confusion as recently as 20 years ago.

Comment author: JoshuaZ 09 June 2010 01:39:49PM 2 points [-]

The problem causes a lot of confusion. There are studies which show that this is in fact cross-cultural. It seems to deeply conflict with a lot of heuristics humans use for working out probability. See Donald Granberg, "Cross-Cultural Comparison of Responses to the Monty Hall Dilemma" Social Behavior and Personality, (1999), 27:4 p 431-448. There are other relevant references in Jason Rosenhouse's book "The Monty Hall Problem." The problem clashes with many common heuristics. It isn't that surprising that some mathematicians have had trouble with it. (Although I do think it is surprising that some of the mathematicians who have had trouble were people like Erdos who was unambiguously first-class)

Comment author: arundelo 09 June 2010 02:36:02PM *  3 points [-]

Erdos

Wow! I looked this up and it turns out it's described in a book I read a long time ago, The Man Who Loved Only Numbers (do a "Search Inside This Book" for "Monty Hall"). Edit: In this book, the phrase "Book proof" refers to a maximally elegant proof, seen as being in "God's Book of Proofs".

I encountered the problem for the first time in a collection of vos Savant's Parade pieces. It was unintuitive of course, but most striking for me was the utter unconvincibility of some of the people who wrote to her.

Comment author: thomblake 09 June 2010 02:41:09PM 2 points [-]

the utter unconvincibility

Yes, my fallback if my intuition on a probability problem seems to fail me is always to code a quick simulation - so far, it's always taken on about a minute to code and run. That anyone bothered to write her a letter, even way back in the 70's, is mind-boggling.

Comment author: AlephNeil 09 June 2010 02:50:45PM 0 points [-]

Yeah it's remarkable isn't it?

I suppose the thing about the Monty-Hall problem which makes it 'difficult' is that there is another agent with more information than you, who gives you a systematically 'biased' account of their information. (There's an element of 'deceitfulness' in other words.)

An analogy: Suppose you had a coin which you knew was either 2/3 biased towards heads or 2/3 biased towards tails, and the bias is actually towards heads. Say there have been 100 coin tosses, and you don't know any of the outcomes but someone else ("Monty") knows them all. Then they can feed you 'biased information' by choosing a sample of the coin tosses in which most outcomes were tails. The analogous confusion would be to ignore this possibility and assume that Monty is 'honestly' telling you everything he knows.

Comment author: MartinB 10 June 2010 01:34:13AM 1 point [-]

Expert confidence. I read vos Savants book with all the letters she got and like how the problem seems to really be a test for the mental clarity and politeness of the actors involved.

Anyone knows of problems that get similarly violent reactions from experts?

Comment author: thomblake 09 June 2010 01:43:43PM 1 point [-]

From Wikipedia:

Monty Hall did open a wrong door to build excitement, but offered a known lesser prize—such as $100 cash—rather than a choice to switch doors. As Monty Hall wrote to Selvin: And if you ever get on my show, the rules hold fast for you—no trading boxes after the selection. (Hall 1975)

The citation is from a letter from Monty himself, available online here.

I'm not sure how the article you linked to is relevant. It does describe an instance of Monty Hall actually performing the experiment, but it was in his home, not on the show.

Comment deleted 09 June 2010 03:25:21PM [-]
Comment author: RobinZ 09 June 2010 04:16:59PM 1 point [-]

thomblake's remark was relevant too, though - from what I said, you might imagine that Monty Hall let people switch on the show. All the clarifications are relevant and good.

Comment author: RobinZ 09 June 2010 11:49:20AM 0 points [-]

Aaargh! And I had upvoted that, believing a random Internet comment over a reliable offline source! That's a little embarrassing.

The article is awesome, by the way. Thanks!

Comment author: thomblake 09 June 2010 01:44:18PM 1 point [-]

See response here