Math Puzzle!

OK kids, here's the deal. An economist who shall remain nameless (but he writes for the NYT) thought he blew up the theory that the recession was due to a misallocation of labor across sectors, because the change in unemployment from Dec-07 to Dec-08 was high in several states that didn't experience a big housing bubble.

I thought that was an odd measure, and so I ranked states (A) by their change in unemployment rates from Jun 06 to Dec 08, and then I ranked the states again by (B) their percentage change in house prices from 2q 06 to 4q 08.

I found that 5 of the top 6 states in column (A) were in the top of column (B). So that seems like a very strong relation.

Can someone talk me through how I would compute the chance that this just happened randomly?

For extra credit, how do we deal with the fact that after I saw the results, I came up with the best way to describe them? In other words, I originally looked at the top 10 states in both lists. But since there were no more matches in the 7th through 10th slots, I just dropped them and reported "5 out of the top 6 match." So I'm thinking this cherry picking after the fact may be contributing to the impressiveness of my result, but maybe not?


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  3. Andy Stedman1:23 PM

    Odds that #1 is in one of top 6 spots: 6/50
    Odds that #2 is in one of remaining 5 top spots: 5/49
    Odds that #3 is in one of remaining 4 top spots: 4/48
    Odds that #4 is in one of remaining 3 top spots: 3/47
    Odds that #5 is in one of remaining 2 top spots: 2/46

    100% x (6/50)(5/49)(4/48)(3/47)(2/46)=0.0002832% (1 in 353,127).

    Generalization: odds that the top A of C values are in the top B of C spots is:

    (B! / (B-A)!) / (C! / (C-A)!)

    So the odds that the top 5 were in the top 10 spots is:

    (10! / 5! ) / (50! / 45!) = 0.0119% (1 in 8408).

    Math checks:
    Odds that the top 1 is in the top spot is (1!/0!) / (50!/49!) = 1/50, as expected.
    Odds that the top 50 are all in the top 50 spots is (50!/0!)/(50!/0!) = 1, as expected.
    (note: 0! is defined to be equal to 1.)

    I'm not promising this is right, but it looks good to me.


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