# Family problems

In an earlier post, I set the following problem:

You’re at a party and meet someone. After chatting for a bit, you work out that the girl you’re talking to has 2 kids and one of them is a boy. What are the chances she’s got 2 boys rather than a boy and a girl?

Following some feedback, I later updated that post to clarify that the question isn’t intended to be about possible slight differences in the birth rates of boys and girls. That’s an interesting biological and demographic issue, but wasn’t intended as the point of the question. For the purposes of the question I simply meant to assume that all children, regardless of a mother’s previous history of births, is equally likely to be a boy or a girl.

In that case, it’s very tempting to answer the question above as 1/2. Indeed, this was the point of the post. One child’s a boy. The other is just as likely to be a boy or a girl, so the answer must be 1/2.

Except it’s not.

The answer is 1/3, and here’s the reasoning…

Without any additional information, if a woman has a 2-child family the possibilities (with the oldest child listed first) are:

Boy- Boy, Boy-Girl, Girl-Boy, Girl-Girl

and because all children are equally likely to be male or female, these combinations are all equally likely. But we can rule out the Girl-Girl combination from the information in the problem, so the remaining possibilities are

Boy-Boy, Boy-Girl, Girl-Boy

with each being equally likely. But if you consider a boy in one of these pairs, it’s only in the first case that the other child is a boy. So, in just 1 of the 3 equally likely outcomes is the other child also a boy, so the probability is 1/3.

This simple problem illustrates the difficulty in calculating what is called a conditional probability – the probability of something conditional on some given information, in this case that one of the children is a boy. Whereas in general the chance of a child being a boy is 1/2, once you include the extra information that we’re looking at a 2-child family in which at least one of the children is a boy, the probability changes. At first it seems counter-intuitive, but once you’ve seen a few problems of this type, your intuition becomes sharper.

With that in mind, let me pose the same problem as above, but suppose you find out that one of the woman’s kids is a boy that was born on a Tuesday. Now what’s the probability that the other child is also a boy?

As usual, I’ll write a future post with the solution and discussion. But if you’d like to send me your own ideas, either by mail or via this survey form, I’d be really happy to hear from you.

Thanks to those of you who replied to the original question. Apart from some initial confusion about whether I was suggesting boys might be more or less common than girls in general, there was roughly a 50-50 split in answers between 1/2 and 1/3. As explained above, it’s easy to be misled into thinking the answer is 1/2, so there is no embarrassment in arriving at that answer. And like I say, that was the whole point of the post. Nonetheless, well done to those of you who got the correct answer of 1/3.

It’ll be interesting to see what replies I get to the revised problem above, so please do send me your answer or thoughts on the problem if you have time.