The sisters "paradox" – counter-intuitive probability

Summary

It seems simple, but it isn'tThere are a couple of famous counter-intuitive problems in probability theory and the sisters "paradox" is one of them. I'll tell you the problem, let you guess the solution, and then give you some of the background.Here's the problem: a family has two children. You're told that at least one of them is a girl. What's the probability both are girls?(International Film Service / American Releasing Co., Public domain, via Wikimedia Commons)Assume that the probability of having a girl or boy is 50% and that the birth order has no effect on the probability. Assume the family is selected at random because they have at least one girl.What do you think the probability is that both children are girls?A simpler questionLet's image you're asked a simpler question.A family has two children. What's the probability both are girls?We can work this out using a simple probability tree:Boy (0.5) Girl (0.5)/ \ / \Boy-Boy (0.25) Boy-Girl (0.25) Girl-Boy (0.25) Girl-Girl (0.25) So the probability of two girls is 0.25.Note there are two ways of having a boy and a girl, so the total probability of having a boy and a girl (in any order) is 0.5.The wrong answerLet's go back to the original problem and see the logic behind the most-often given wrong answer.The birth chance is 0.5 boy and 0.5 girl. We don't know the gender of one of the children, but it must be a 0.5 probability it's a girl. Given the fact we already know one of the children is a girl, the probability of their being two girls must therefore be 0.5.It sounds right because it sounds logical, but it isn't right for reasons as I'll explain next.The correct answerThe correct answer is 1/3. Let's see why.In the probability tree above, we can see four equally likely combinations: {Boy-Boy} (0.25), {Boy-Girl} (0.25), {Girl-Boy} (0.25), and {Girl-Girl} (0.25). We're told in the problem that the {Boy-Boy} combination is ruled out, which leaves us with three remaining combinations. Each of these three remain...