Two Daughters

A family with 2 children moves in next door. You notice one of the children and observe that it’s a girl. What is the probability that the other child is a girl?

You might like to ponder a while.

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In February 2015 Mathematics Today it is observed that this puzzle is linguistically similar to another one in which you

 

  1. observe the a priori possibilities for the combinations BB, BG, GB, GG
    (where the characters are B for a boy, G for a girl, with the first place for the first born, second for the second)
  2. delete the possibility BB, and
  3. note that only one of the three remaining possibilities has two girls.

But is this right for the question as stated?

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Well, obviously not (or it would not have merited an article).

[As] written above, the answer is 1/2, assuming that nature is random and might equally have chosen to ‘show’ you a boy.

[One has, applying Bayes’ rule] P(GG|G) = P(G|GG).P(GG)/P(G)
= (1 x 1/4)/(1/2) = 1/2.

But is the assumption justified? Is it necessary? It seems to me that this is till answering a different question, it is just that the second suggested answer is more ‘open transparent and honest’ and at least would be right in some possible world. My critique follows.

The line above in which Bayes’ rule is applied has three terms on the right-hand side. The first simply says that if the family has two girls then the unobserved child is a girl. This seems harmless. The other two terms are where the assumptions come into effect. They are:

  • That the genders of children within families are equal and independent.
  • That in a family with a boy and a girl, the boy and the girl are equally likely to be noticed.

It does not seem to me that a mathematician, as such, is the right person to make such judgments. I gather that the first is commonly assumed in schools, but would have more faith in census studies. The second seems highly dubious, unless some extensive experiments have been carried out. In my limited experience, relatively few families with two children have one of each. It is much more common to have two girls or two boys. In this case, P(GG) > 1/4. It also seems to me that I would be more likely to notice children out of doors, and that boys would likely to be more adventurous, move quicker and make more noise, and hence in a mixed family I would be more likely to see the boy first. Hence P(G) < 1/2. Overall, then P(GG|G) > 1/2.

What’s going on?

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The first thing to note is that, as so often, the question is ambivalent about whether it wants an ‘objective’ or ‘subjective’ probability. If, for me, P(GG) > 1/4 and P(G) < 1/2 then ‘for me’ P(GG|G) > 1/2: I am correct. Not being an expert in the necessary domain, I can hardly be expected to work out the objective probability. It seems to me that what is being asked for is neither my subjective probability or some objective probability but some sort of official school text-book multi-subjective probability. The role of a mathematician is then two-fold: to help calculate the official answer, and to help people understand its limitations. Importantly – it seems to me – the ‘official’ answer is not ‘the’ mathematical answer: it is merely that which is commonly taught, at least in the UK.

See Also

Other puzzles.

My notes on probability et al.

Dave Marsay 

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