Consider the following excerpt from a recent article in the British Medical Journal:
Mike has only two children, and they are called Pat and Alex, which could equally be boys’ or girls’ names. In fact, Pat is a girl. What is the probability that Alex is a boy?
b Slightly less than 50%
c Slightly more than 50%
d Between 60% and 70%
e Between 40% and 30%
d—Although this could be about the relative popularity of ambiguous names for boys and girls or about subtle imbalances in the sex ratio, it is not meant to be. The clue to the correct answer is in thinking about what we do not know about the family and what we do know already, and applying this to the expected probabilities of girls and boys.
We do not know if Pat was born first or second. We do know that there are only two children and that Pat is a girl. I am assuming that in the population, 50% of children are girls.
The birth order and relative frequency of two child families are: boy, boy (25%), girl, girl (25%), boy, girl (25%) girl, boy (25%). We know Mike’s family does not have two boys, since Pat is a girl, so we are only left with three choices for families with at least one girl. Two of these pairs have a boy and one does not. Hence the probability that Alex is a boy is two thirds or 66.7%.
If we had been told that Pat was born first then the probability that Alex is a boy drops to 50%.
The well-known "Boy or Girl" paradox that is referenced in the fragment above is probably as old as the probability theory itself. And it is therefore quite amusing to see an incorrect explanation for it presented in a serious journal article. You are welcome to figure out the mistake yourself.
For completeness sake, here is my favourite way of presenting this puzzle:
In the following, let Mike be a randomly chosen father of two kids.
- Mike has two kids, one of them is a boy. What is the probability that the other one is a girl?
- Mike has two kids, the elder one is a boy. What is the probability that the other one is a girl?
- Mike has two kids. One of them is a boy named John. What is the probability that the other one is a girl?
- I came to visit Mike. One of his two kids, a boy, opened the door to me. What is the probability that Mike's other child is a girl?
- I have a brother. What is the probability that I am a girl?
- I have a brother named John. What is the probability that I am a girl?
You can assume that boys and girls are equiprobable, the births of two kids are independent events, a randomly chosen boy will be named John with probability p, and that a family may have two kids with the same name.
If you haven't tried solving these yet, give it a try. I'm pretty sure you won't do all 6 questions correctly on the first shot.