Intuition is a great asset, but:

- Can sometimes lead you astray;
- Always needs to be balanced by doubt, to avoid becoming complacency;
- Is never a substitute for genuine understanding and knowledge.

A while back I posed the question

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?

And I discussed the solution to this problem in a subsequent post. As I discussed, although it’s tempting to give an answer of 1/2 – arguing that the other child is equally likely to be a boy or a girl – this reasoning is wrong, and the correct answer is 1/3.

I then followed up this discussion by extending the original problem as follows. Everything is the same, except you are told that the woman has 2 children, one of whom is a boy that was **born on a Tuesday**. With this information, what is the probability that the other child is also a boy?

Again, it looks like there ought to be an easy answer to this problem. And again, it turns out that this easy answer is wrong. But this time the breakdown of the simple intuitive argument is often surprising even to people who work regularly with probabilities.

The intuitive – but wrong – answer is that the probability is still 1/3. The argument goes that there is no relationship between gender and day of birth. Technically, gender and day of birth are statistically independent. So, the information provided by the day of birth is completely irrelevant and can be ignored, meaning the probability that the other child is a boy remains at 1/3.

Except it doesn’t. The true answer is 13/27, which is just slightly less than 1/2.

The calculation leading to this answer is not difficult, but slightly more complicated than is reasonable for me to include in this post. If you’re interested, you can find a full explanation here.

But, as explained in that article, it’s interesting to state the result in slightly more generality. We added the condition that the boy was born on a Tuesday, an event which has probability 1/7 for both boys or girls. Suppose we replace this event with something else like ‘The boy is left-handed’; or ‘The boy has green eyes’; or ‘The boy was born on Christmas Day’. And suppose the probability of a child – boy or girl – fulfilling this extra condition is p.

For the original Tuesday’s child problem, p=1/7. For the Christmas birthday p=1/365. For green eyes it’s whatever the proportion of people in the population with green eyes is. And so on.

Anyway, if we replace the condition of being born on a Tuesday with some other condition whose probability is p for either boys or girls, it turns out that the probability that both the children are boys, given the information provided, is

If we substitute p=1/7 in this expression we get Q = 13/27, which explains the answer given above. But it’s not the value itself that’s important; it’s the fact that it’s different from 1/3. So, although gender and day of birth are independent, giving the extra information about day of birth shifts the original probability from 1/3 to 13/27.

And there’s something even more interesting. If the day of birth extra condition is replaced with a condition that’s less likely, so that p is smaller, then the value of Q gets closer and closer to 1/2. For example, with the Christmas birthday condition, p=1/365 leading to Q = 0.4996573. In other words, if we include a very unusual condition for the child known to be a boy, the probability that the other child is also a boy gets very close to 1/2, which is the answer you get to the original problem by using the wrong intuition.

Not very intuitive, but it’s the truth.

Finally, here’s Dilbert’s take on intuition:

I hate to come late to the party, but the answer to the original question is 1/2. And no, it isn’t because the chances that “the other child” is a boy are 1/2.

Before I explain why, recall the Monty Hall Problem (I assume you know it). Most people say that there were three cases originally, one was eliminated, so each that remains has a 1/2 chance. Compare this to your solution here: there were three cases originally, one was eliminated, so each that remains has 1/3 chance. This is the same solution, and it is wrong both times. It is wrong because probability is not concerned with just the outcomes that look like they could have happened, it is concerned with how they could happen. And some that look right could not.

The case count is correct in both solutions. But you can’t count them at full probability. In the MHP, if you picked wrong originally, then Monty Hall has only one choice of a door he can open. So there is a 1/3 chance of him opening the door he did *AND* your choice being right, as most people think. But if you picked wrong he had two choices. There is a 1/6 chance of him opening that door *AND* your choice being wrong, not the 1/3 chance people think.

In your problem, if the girl you are talking two has a boy and a girl, and from the conversation you work out one of the genders? There is a 1/2 chance that you “work out” that one is a boy, and a 1/2 chance that you “work out” that one is a girl. So the probabilities you should use are 1/4 for boy-boy, and 1/8 each for boy-girl and girl-boy. The chances she has two boys are 1/2.

And now the intuition you talk about – that the information “born on a Tuesday” is irrelevant – gives the right answer. It is 1/2 either way.

Oops; I meant that in your problem, there originally were four cases.