December 31, 2008Finishing The GameIn yesterday's post, I asked this 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? Most people answer 50%.Unfortunately, this isn't correct.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. This is a fancypants way of saying there were almost a thousand comments on that post, with not a lot of agreement to be found.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. We can use that information to come up with a better, more correct answer. We know this person has two children. What are all possible combinations of two children?BB, GB, BG, GGtwo children possibility matrix: BB, GB, BG, GGWe know that one of the children is a girl. This rules out one of those possible combinations of two children (BB), so we're left with:GB, BG, GGtwo children possibility matrix, has a girl: GB, BG, GGOf the remaining three possibilities, two include boys.GB, BGThus, the odds of this person having a boy and a girl is 2/3 or 66%.I noticed a few comments where people complained that the GB and BG possibilities are the same thing, and should have been reduced toBG/GB, GGWhich equates to 1/2 or 50%.
