On probabilism and statistics

8 February, 2013 at 12:45 | Posted in Statistics & Econometrics | 3 Comments

‘Mr Brown has exactly two children. At least one of them is a boy. What is the probability that the other is a girl?’ What could be simpler than that? After all, the other child either is or is not a girl. I regularly use this example on the statistics courses I give to life scientistsworking in the pharmaceutical industry. They all agree that the probability is one-half.

dicing-with-death-chance-risk-and-healthSo they are all wrong. I haven’t said that the older child is a boy.The child I mentioned, the boy, could be the older or the younger child. This means that Mr Brown can have one of three possible combinations of two children: both boys, elder boy and younger girl, elder girl and younger boy, the fourth combination of two girls being excluded by what I have stated. But of the three combinations, in two cases the other child is a girl so that the requisite probability is 2/3 …

This example is typical of many simple paradoxes in probability: the answer is easy to explain but nobody believes the explanation. However, the solution I have given is correct.

Or is it? That was spoken like a probabilist. A probabilist is a sort of mathematician. He or she deals with artificial examples and logical connections but feel no obligation to say anything about the real world. My demonstration, however, relied on the assumption that the three combinations boy–boy, boy–girl and girl–boy are equally likely and this may not be true. The difference between a statistician and a probabilist is that the latter will define the problem so that this is true, whereas the former will consider whether it is true and obtain data to test its truth.

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3 Comments

  1. The answer surely depends on how you found out that “At least one of them is a boy”
    Did Mr Brown volunteer the information? Did you see 1 of his children and it happened to be a boy? Did you ask him “Is at least one of your children a boy?”
    The answer can be either 1/2 or 2/3.The way the problem is usually posed the most reasonable answer is 1/2.

  2. Hey Lars, I responded to one of your papers. I think that there’s an interesting debate to be had here. I really like your work — even if I think we can and should take a slightly different approach:

    http://fixingtheeconomists.wordpress.com/2013/07/10/economics-as-social-organisation-why-we-should-all-be-relativists/

    Phil

    • Interesting reading Philip. I’ve written a lot on social constructivism and postmodern methods within social sciences, but so far only in Swedish.
      I’ll try to get some time this summer and try to formulate my main points of critique in the internationally more accessible English language.
      Appreciate you taking your time relating your own thoughts to my papers!


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