Logic fail…?

Jeff Atwood posed a question,

Let’s say, hypothetically speaking, you met someone who told you they had two children, and one of them is a girl. What are the odds that person has a boy and a girl?

What’s your answer?

Jeff’s posts on the question generated thousands of comments. Many, many people simply refuse to accept the answer. The correct answer is that the odds the person has a boy and a girl is 67% (two-thirds).Most people argue strenuously that the correct answer is 50%. After all, the sex of one child doesn’t affect the sex of the other child does it?

Jeff poses the question as an illustration of two things: our inability to process probability and our inattention to detail,

This problem, although seemingly simple, is hard to understand. For cognitive reasons that are not fully understood, while our intuitions regarding a priori possibilities are fairly good, we are easily misled when we try to use probability to quantify our knowledge….

The key thing to bear in mind here is that we have been given additional information. If we don’t use that information, we arrive at 50% — the odds of a girl or boy being born to any given pregnant woman. That’s true insofar as it goes, but it’s the answer to a different, much simpler question, and certainly not the answer to the question we asked.

Our question contains additional information:

  1. The person has two children.
  2. One of those children is a girl.

But rather than pose additional information, the question hides information. The question as it’s usually asked masks the nature of the problem and relies on the listener’s natural inclination to analyze problems in context and arrive at a conclusion that makes sense in the real world.

The solution to the problem runs like this:

Say there are 100 couples. They each have a child. 50 will have boys and 50 will have girls. Each couple then has another child. Of the 50 that had boys, 25 will have girls and 25 will have boys. Of the 50 that had girls, 25 will have boys and 25 will have girls. The final breakdown will be: 25 each of Bg, Bb, Gg, Gb (uppercase indicates the older child).

If we know that the person has a girl, that leaves only three possible child combinations: Gb, Gg, Bg. Two of those combinations (67%) have a boy.

If we ask the question, what are the odds that any given couple will have a boy and a girl, the answer is 50%. (Bg, Gb) If we ask what percentage will have a boy, the answer is 75% (Bg, Bb, Gb). If we’re told that one child is a girl, that removes 25 couples from our pool (the 25 Bb couples have no girls). If we ask, of those remaining (Gg, Bg, Gb), how many have boys, the answer is 67% (Bg, Gb).

“But!” Say the doubters, “Shouldn’t we eliminate the artificial distinction about which child came first? Gb and Bg are functionally equivalent! The combinations for gender are: BB, GG, and BG. We know the person has a girl, which leaves GB and GG, so 50%! Birth order does not matter!”

This argument rages in Jeff’s comments, and as Jeff points out, that’s the answer to a different question. That’s the answer to, “A person has a child, that child is a girl. What are the odds that his next child will be a boy?”

Sure, that’s not the question that was asked, and so we can say that many people don’t listen. But that’s not really fair. The question is kind of silly. It’s a pointless abstraction that pretends to divine the listener’s mathematical intuition, but it does that by deliberately obfuscating the problem.

For example, we could phrase the question this way:

“Suppose you meet someone. This person tells you that he has flipped two coins. What are the odds the the person has both heads and tails? What are the odds that the person both heads and tails if we know that the person does not have two heads?”

It’s not phrased that way because that makes it look like a logic problem. The question as asked is designed to elicit “incorrect” answers. It does that by relying on the ability of the listener to make sense of the world.

If we ask the question as first given, the average listener places the question into a familiar, real-world context and imagines a conversation with another person. In the real world, if someone says to you, “I have two children and one of them is a girl.” The natural assumption is that the other child is a boy because why else would the person phrase it that way? If he had two girls he should have said “I have two children, both girls.” If he does have two girls but still says, “one of them is a girl” then he’s being deliberately evasive.

Why does it matter? It matters because context matters. The point of logic is to make sense of the world. If we’re writing a computer gambling program, then we need to be sure that we understand probability distributions and game theory. In that context, abstraction is appropriate. If we’re having a conversation in the lunch-room at work, then we need to emphasize a different set of essentials.

No problem exists as pure abstraction, there’s always a context and that context is always relevant. Whether that context is a deck of cards, a set of coins, a lunch-room conversation, or your neighbor’s money, we can’t solve real-world problems by ignoring context.