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

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

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Comment author: ciphergoth 03 June 2010 10:14:07AM *  15 points [-]

I always much prefer these stated as questions - you stop someone and say "Do you have exactly two children? Is at least one of them a boy born on a Tuesday?" and they say "yes". Otherwise you get into wondering what the probability they'd say such a strange thing given various family setups might be, which isn't precisely defined enough...

Comment author: Emile 03 June 2010 01:17:35PM *  4 points [-]

Very true. The article DanielVarga linked to says:

If you have two children, and one is a boy, then the probability of having two boys is significantly different if you supply the extra information that the boy was born on a Tuesday. Don't believe me? We'll get to the answer later.

... which is just wrong: whether it is different depends on how the information was obtained. If it was:

-- On which day of the week was your youngest boy born ?

-- On a Tuesday.

... then there's zero new information, so the probability stays the same, 1/3rd.

(ETA: actually, to be closer to the original problem, it should be "Select one of your sons at random and tell me the day he was born", but the result is the same.)

Comment author: Christian_Szegedy 04 June 2010 12:09:44AM *  1 point [-]

I think the only reasonable interpretation of the text is clear since otherwise other standard problems would be ambiguous as well:

"What is probability that a person's random coin toss is tails?"

It does not matter whether you get the information from an experimenter by asking "Tell me the result of your flip!" or "Did you get tails?". You just have to stick to the original text (tails) when you evaluate the answer in either case.

[[EDIT] I think I misinterpreted your comment. I agree that Daniel's introduction was ambiguous for the reasons you have given.

Still the wording "I have two children, and at least one of them is a boy-born-on-a-Tuesday." he has given clarifies it (and makes it well defined under the standard assumptions of indifference).

Comment author: DanielVarga 04 June 2010 08:33:00AM *  6 points [-]

Yesterday I told the problem to a smart non-math-geek friend, and he totally couldn't relate to this "only reasonable interpretation". He completely understood the argument leading to 13/27, but just couldn't understand why do we assume that the presenter is a randomly chosen member of the population he claims himself to be a member of. That sounded like a completely baseless assumption to him, that leads to factually incorrect results. He even understood that assuming it is our only choice if we want to get a well-defined math problem, and it is the only way to utilize all the information presented to us in the puzzle. But all this was not enough to convince him that he should assume something so stupid.

Comment author: Christian_Szegedy 04 June 2010 06:03:52PM 3 points [-]

For me, the eye opener was this outstanding paper by E.T. Jaynes:

http://bayes.wustl.edu/etj/articles/well.pdf

IMO this describes the essence of the difference between the Bayesian and frequentist philosophy way better than any amount of colorful polygons. ;)

Comment author: ciphergoth 03 June 2010 02:27:58PM 1 point [-]

I get that assuming that genders and days of the week are equiprobable, of all the people with exactly two children, at least one of whom is a boy born on a Tuesday, 13/27 have two boys.

Comment author: Emile 03 June 2010 03:53:05PM *  1 point [-]

True, but if you go around asking people-with-two-chidren-at-least-one-of-which-is-a-boy "Select one of your sons at random, and tell me the day of the week on which he was born", among those who answer "Tuesday", one-third will have two boys.

(for a sufficiently large set of people-with-two-chidren-at-least-one-of-which-is-a-boy who answer your question instead of giving you a weird look)

I'm just saying that the article used an imprecise formulation, that could be interpreted in different ways - especially the bit "if you supply the extra information that the boy was born on a Tuesday", which is why asking questions the way you did is better.