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Mathematicians don't understand people? [Dec. 31st, 2008|01:47 pm]
Arvind Narayanan
 [ Tags | math, probability, puzzle ]

This is again a comment that I posted elsewhere, in response to the following question:
Someone tells you they have two children, and one of them is a girl. What are the odds that person has a boy and a girl?
My answer, reproduced below, got voted up like crazy—I guess it struck a chord with nearly everyone. I know that many of you are math people.. I look forward to your vehement objections :-)
This is a famous puzzle. The answer is supposed to be 2/3, because what the question is asking you to do is consider all the parents in the world where at least one of the two children is a girl. Then you're left with 3 possibilities, BG, GB and GG.

If you phrase the question like that, everyone will get the right answer. The reason people get it wrong is that people don't normally talk like that. Imagine you're at a party, and someone tells you they have two kids, and "one of them is a girl." Clearly, they mean that the other is a boy, which means the answer is 100%.

But the most intuitive way of interpreting the question is that you know that a specific child is a girl, say because the person brought one kid to the party, who turns out to be a girl. With this interpretation, the obvious answer of 50% is in fact correct.

You often hear the complaint that people don't understand math. In this instance, however, an equally valid way of explaining what's going on is that mathematicians don't understand people.

This criticism applies partially to the normal game-show version of the Monty Hall problem, but I think there the wording is genuinely ambiguous regarding the host's behavior, and my answer would be "not enough information."
Seriously, when is the last time anyone told you that at least one of their kids was a girl? You could imply it, for instance by saying "I need to pick up my daughter from school," but I cannot think of a phrasing in English that lets you state it directly. Not without sounding like a huge dork, anyway.

Edit. For the record, the missing information in the Monty Hall problem that I'm talking about is, "does the host only open doors which he knows not to have the prize, or any door other than the one the contestant picked?"

 From: 2008-12-31 08:36 pm (UTC) (Link)
A large part of the problem people have with word problems is translating from the words into a valid formula. Once they have the valid formula, they proceed better, even if they are not whizzes.

And there is a big difference between speaking colloquially and speaking precisely. I suppose the people who appreciate the difference are those "who don't understand people". :-)

There is need, at times, to speak precisely. One can't have technical discussions totally in colloquial language - there would be too much ambiguity.

But most people, most of the time, communicate colloquially just fine. And don't worry about the exceptions. Except when someone asks them about odd probabilities. :-)

 From: 2008-12-31 10:48 pm (UTC) (Link)
Oh hi, I was reading friends of friends and found your journal.

Thanks for expressing something that had always bothered me. The common so-called fallacies, it seems to me, are products of the assumptions made in simple human communication. One must assume an interlocutor who is trying to make sense and will provide you with relevant detail.

The only time this really goes wrong is when we apply this logic to the data we get from machines, or statistics. That's where we humans, who assume an interlocutor, can mistake the data as "saying" something. Then we have to be extra careful.

Edited at 2008-12-31 10:48 pm (UTC)
 From: 2009-01-01 11:10 pm (UTC) (Link)
cool. i didn't even know there was a friends-of-friends!

do you do security? you seem to know many security folks. work for flickr, huh? awesome. it'd be nice to get coffee next time i'm there.

 From: 2009-01-02 08:09 am (UTC) (Link)
i didn't even know there was a friends-of-friends!

Try http://arvindn.livejournal.com/friendsfriends .

do you do security? you seem to know many security folks.

No more than any other web geek ought to know.

I'm not sure that I do know lots of security people. There's but I know him through entirely non-computer channels.

Forgive me if i don't friend you back; I did subscribe in Google Reader but I tend to use friends-only LJ for more personal stuff. But, it might be interesting to sync up sometime. Look me up if you're in the Bay Area. I'm definitely interested in the "...and what to do about it" part of your research.
 From: 2009-01-02 04:27 pm (UTC) (Link)
there's also and .