Answer to Puzzle #45: If one child is a boy, what is the probability the other is a boy?

45. A lady has two children. One is a boy. What are the chances of the other child also being a boy?

How does this change if you are told the oldest child is a boy?

A classic, lets take a look:

Before reading the answer can I interest you in a clue?

As experienced puzzlers we read this a bit differently than most people. What it does not say is only one child is a boy. The wording of the question deliberately doesn't exclude two children being boys by saying one child is a boy. The other thing we should probably pick up on as experienced people in this field is that the answer, however intuitively obvious it may be, is not going to be 50%.

The table below shows the possible gender combinations of two children.

  Boy Girl
Boy BB GB
Girl BG GG


It should now be clear that the green cells match the criteria of the question, that (at least,) one child is a boy. There are three of them. BB, GB and BG. In only one of them is the 'other child' a boy. So the probability is one third.

If you're still not convinced then I have further proof hidden below click here to view it...

The Eldest Child is a Boy

In a sense the problem becomes a lot simpler at this point. In the first part it is a little counter intuitive to declare that there is a 33% of what appears to be an individual child being a boy. Of course it is not an individual child, it's a combination of the probabilities of two children. In this case however, we are essentially being asked the probability that a specific child is a boy and this is clearly 50%.

It's probably not necessary to demonstrate, but for completeness...

    Eldest Child
    Boy Girl
Youngest
Child
Boy BB GB
Girl BG GG


As if it were needed we have shown that there are two cases in which the eldest child is a boy, and in only one of them is the youngest child a boy. The answer is 50%.

Assumptions

I've been meaning to be more rigorous about explicitly stating our assumptions. It would be legitimate for an interviewer to ask and challenge you over what assumptions you are making. Sometimes they are so obvious they don't even seem like they are up for debate.

In this case we are assuming that the gender of two siblings are uncorrelated independent variables with a 50% distribution.

Challenges to this of varying merit might be that for whatever reason some people may have a higher propensity to give birth to one sex or the other. So the existence of one boy may make another more likely. This is a fairly weak point. A better one might be that we know identical twins exist. And in whatever proportions of births they make up we can say for certain that all other things being equal this will increase the chances of two siblings having the same gender.

To be clear I am not saying that this affects the answer. Just that it's worth knowing the assumptions you are making.





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