We came across the ‘two girls’ version of the children's gender problem nearly 35 years ago. How we came to it we cannot remember, but Martin Gardner had published a variant of it in the Scientific American in 1959. It re-emerged for us in the summer of 2010, following the publication of an article in Science News [1]. Subsequently Keith Devlin wrote about how this re-emergence impacted on him, and noting that ‘Probability Can Bite“ [2]. The mathematics herein reflects and extends that in Devlin's article.
In case the reader has not encountered the problem before, we first pose four problems.
1. A family has two children. One of them is a girl. What is the probability that they are both girls?
2. A family has two children. The younger is a girl. What is the probability that they are both girls?
3. A family has two children. One of them is a girl, and she was born on a Tuesday. What is the probability that they are both girls?
4. A family has two children. One of them is a girl, and she has green hair. What is the probability that they are both girls?