two kids

Here's another interesting brain teaser that has been making the rounds. It creates the same kind of exasperation in people that the "Monty Hall problem" and the "airplane on a treadmill" problems created.

Here's the problem: You meet someone who has two kids. They tell you that one of them is a girl. What are the odds that the other kid is a boy?

It's pretty innocuous sounding but reading some of the commentary on other blogs it elicits the kind of aggression you'd expect from UFC fighters. There is the usual discussion of how the problem is worded poorly and people taking strong stands on upwards of 4 different answers.

The interesting thing about this problem isn't that people get it wrong. It's as confusing and bewildering as those other two problems. It is how many people are just not interested in learning something from it. And by this I don't just mean the people who get it wrong. The people who get the right answer inevitably go on to explain the answer in ways that are completely wrong. Everyone seems to take an attitude that the other side consists of a bunch of intellectual retards and how can they possibly not see why they are stupid. It's mainly about bragging rights and self affirmation.

As I've written before this is exactly why research papers, even in peer reviewed journals, must be viewed skeptically. It's a natural tendency for people to arrive at a conclusion (regardless of whether the answer is right or wrong and regardless of whether the conclusion was arrived at with good or bad methods) and then stick to it. We call it "anchoring" in my business. You definitely see this in situations where there is too much data to really do all the analysis such as in stock picking or topics like global warming. I came to the 50% answer originally but when the answer was stated as 66% and I didn't understand the explanation I tried to let go of my answer. It's a hard thing to do. Unfortunately the explanation is typically wrong or ambiguous which made me dig a little deeper into what was going on.

Let's get to the answer: There is a 66% chance the kid is a boy.

Here are the 3 most common wrong answers and reasoning.

100% Chance.
If the other kid was a girl then the other must be a boy otherwise the person would have said they had two girls. QED.

50% Chance.
We know one kid is a girl so the other must be a girl or a boy. The odds of a girl is 50% and the odds of a boy is 50%. QED.

66% Chance.
There are 4 possible outcomes for someone with two kids. Girl-Girl, Girl-Boy, Boy-Girl, Boy-Boy. If we know there is at least one girl then there are three possible outcomes (GG, GB, BG). Two of those three outcomes has a boy so there is a 2/3 or 66% chance the other kid is a boy. QED.

100% is a classic misreading of the problem. It makes the assumption that the person would have said there are two girls if the other kid was a girl. But the problem doesn't say that so you cannot assume it. Obviously wrong.

50% is the intuitive answer. The reasoning is almost correct. In some ways it is more correct than how the reasoning is presented by people who answer correctly. It's hard to find the flaw.

66% is correct but people who don't agree seem to implicitly not understand why GB and BG are treated as separate outcomes. Why are they different? In some sense the "50%-ers" are right in that they aren't different on a certain level.

For me the insight was working through the problem in a slightly different way that combines both approaches. The GB and BG outcomes are not different in some sense. But the probability that both a boy and a girl is produced is different from the odds that a girl-girl or boy-boy is produced. Any couple with two kids has a 50% chance of having mixed genders and a 25% chance of having two girls and a 25% chance of having two boys. This should be easy to work out.

If we know that at least one of them is a girl then our possible solution must be from mixed gender (50% probability) or two girls (25% probability). So when we answer the question what is the probability that the other kids is a boy we have 50%/(50% + 25%) or 66%.