On probabilism and statistics

12 Jun, 2022 at 21:22 | Posted in Statistics & Econometrics | 6 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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  1. The 1 in 3 answer assumes that there is a difference between younger girl/older boy and older girl/younger boy; yet the question has nothing to do with who is older or younger – he order of birth is irrelevant. They could be twins for example.

  2. What an odd answer. Literally. He excludes the two girl case, extends conditioning age information on the possible sister, but then fails to do the same for the possible brother by not recognizing that the brother could also be either older or younger. That would put things back to even. Maybe the scientists know something after all.

  3. You haven’t said what the children’s hair color is either, but you could make the same sort of argument. The reason the probability changes from 1/2 to 2/3 is that they are answers to different questions.

  4. Karey’s reference is very relevant. As a mathematician, I would hope that any such example would be given in a context where both the assumptions and intended interpretation were clear. It seems to me that some probabilists are better at this than others. On the other hand, not all statisticians are as good as I would wish, either.

    The ‘common sense’ interpretation of ‘p=2/3’ is that it is a reasonable approximation as long as one makes certain ‘standard assumptions’. My problem is that these never seem to apply to any actual situations in which I am faced with statements like those quoted. In particular, economics seems riddled with assumptions that seem much less tenable than those of Senn’s probabilist, and it seems to matter.

    If only we had a clear and accessible general account of the issues, we could maybe move on? (I have just updated my own attempt at https://djmarsay.wordpress.com/notes/puzzles/the-two-daughter-problem/ , where I would welcome comments. )

  5. Can’t kids choose their own gender these days?

  6. Nice example of the former here: ‘The[re is a] slight excess of male births’ so the prior probability of male and female is not equal. More like 1:1.05
    It doesn’t look like the sex of the second child depends on the sex of the first, except in the case of two-child families, in which case the is a slight increase in the likelihood the second child will be the opposite sex.
    https://royalsocietypublishing.org/doi/10.1098/rspb.2019.2849


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