p (A|B) != p (B|A)

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A common mistake in conditional probability is to confuse the conditioning (that is, to mistake p(A|B) for p(B|A)). One complication here is that our language for probability can be ambiguous. For example, I have done a classroom demo replicating the experiment of Kahneman and Tversky in which students guess “the percentage of African countries in the United Nations.” I always thought this meant
100*(# African countries in U.N.)/(# countries in U.N.).
But some students thought this meant
100*(# African countries in U.N.)/(# countries in Africa).
So, to even ask the question clearly, I need to ask for “the percentage of countries in the U.N. that are in Africa,” or something like that.

Anyway, I recently went to a talk by Maryanne Schretzman (Dept of Homeless Services, NYC), where an interesting example arose of the difference between p(A|B) and p(B|A). They’re looking at new admissions to the shelter system, and a lot of them come are people who are released from jail. But the jail administrators aren’t so interested in talking about this, because, of all the people released from jail, only a small percentage go to homeless shelters. p(A|B) is high, but p(B|A) is small. Same numerators, but the denominator is much bigger in the latter case.

4 thoughts on “p (A|B) != p (B|A)”

  1. When I taught the African countries in the U.N. trick, I knew that the first question somebody would ask would be `OK, so what is the _correct_ percentage of African countries in the UN?' K&T 1974 is everybody's citation for this, but it doesn't give the correct percentage. In fact, maybe I'm just a statistic-seeking dunce, but I couldn't find a single paper which cited K&T that reported the correct figure.

    So I went to the UN web site and counted myself. I don't know what it was in 1974, but the answer today is 23%.

  3. I have taught this topic as well. I use the example argument that "marijuana is a gateway drug" and the argument is that 90% of all heroin users smoked marijuana first" P(A|B). But if you look at all the marijuana users that eventually try heroin, its like 2% P(B|A). It drives the point home and keeps their attention.

  4. I like Chris' example a lot. Another one I made up last year was:

    p(Saddam Hussein would refuse UN weapon inspectors/hiding WMD)=1

    But p(hidingWMD/refuse inspectors) is much lower than 1.

    The mea culpa editorial that the New York Times published last year apologizing for not being critical enough of the Bush Administration's justification for going to war essentially admitted that the paper failed to keep this distinction straight.

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