The information in the Tuesday's child puzzle

Further to my previous post on 'Tuesday's child'.

In the process of composing a letter to New Scientist about this - see below - I've realised where the extra information is coming from.

In the terms I've used in the letter (I've managed to add confusion by changing who is 'me' and who is 'you' between the letter and my blog post!), the probability that you have two boys increases when you answer 'yes' to my question 'do you have a boy that was born on a Tuesday'. Notice that that question is about both your children, whereas if you had said 'I have a boy that was born on a Tuesday' you are only giving me information about one child.

OK, I've spent far too long on this already.
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Dear Sir/Madam,

I fear you have not got this one (Tuesday's child, letters 4th July) sorted yet. Probabilities unearth lots of counter-intuitive and surprising results, and there's one in here to do with specifying the birth day, but you have to be very, very careful how you present it. I don't think Bellos got it right in the initial article (Mathemagical, 29th May) and the letters of 4th July don't help.

The claim is that adding the day your son was born to the fact that you have two children and one of them is a boy, changes the probability that you have two sons from 1/3 to almost 1/2. However, since your son must have been born on some day, stating what day that is does not by itself change the probabilities. Externally specifying the day, on the other hand, does make a difference.

If you say you have two children and that at least one is a boy then I can conclude that the probability you have two boys is 1/3 (it is more likely that you are one of the people with a boy and a girl) - we're all agreed on that bit of the argument. If you also tell me the day the boy was born nothing changes because I knew all along that he must have been born on one day of the week. If, however, instead of you telling me up-front the day your boy is born, I say 'have you a boy that was born on a Tuesday?' and you say 'yes', that does raise the probability that you have two boys to nearly a half.

The reason it changes in the latter case is that, when I specify a day, someone with two boys is more likely to have one of them born on the day of my choice than someone who only has one boy.

I've teased this out in tedious detail on my blog: intropy.co.uk

Regards,
David Chapman