The “boy or girl paradox” is a well-known brain teaser by famous puzzle-maker Martin Gardner. It’s popularly considered a “paradox” because (1) it has a highly unintuitive solution, and (2) its ambiguous wording meant either of two solutions could be valid solutions.
This is a rewording of that brain teaser to eliminate some ambiguity from that original question:
Out of all families with exactly two children, we randomly pick one family that has at least one boy. What is the probability that both children in this family are boys?
Assume only for the purposes of this puzzle that a child can only be a boy or a girl, and that either possibility is equally likely.
Spoiler warning: solution is below, don’t scroll down if you don’t want to see the solution
Solution to the Boy or Girl Paradox
1 / 3
You might have thought the answer would be 1 / 2. After all, if you know the first child is a boy, then there is a 1 / 2 chance the second child is a boy. This would be correct if we randomly selected a family, and then observed that at least one of the children was a boy.
However, that was not the case here. The question states that a family was randomly selected on the basis of having at least one boy. That changes the likelihood that we chose a family with two boys.
There are 4 equally likely arrangements for a family with two children:
If we randomly choose a family that has at least one boy, that eliminates the first possibility, leaving three equally likely possibilities. Out of those three, only one has two boys. Therefore the probability that both children in this family are boys is 1 / 3.
Common errors and fallacies
But older and younger shouldn’t matter. It’s either boy-girl or boy-boy, so the probability should be 1 / 2.
Older and younger don’t matter, it’s just a way of distinguishing between the two children so it’s easy to visualize in a table (you could instead number them child 1 and child 2). However, boy-girl and boy-boy are not equally likely! Boy-girl is twice as likely as boy-boy, so the probability remains 1 /3.
Think of this as coin flips instead: the probability of flipping two coins and getting 1 heads and 1 tails is twice as likely as getting 2 tails. That’s because there are two equally likely ways of getting 1 heads and 1 tails (first heads then tails, or first tails then heads), but only one way of getting 2 tails.
How the family was selected shouldn’t matter. We know one child is a boy, so the remaining child must be equally likely a boy or girl.
How something is selected always matters in probability. The key is we don’t know that one specific child is a boy, we know that a family with at least one boy is chosen—that boy could be either child. Because it could be either child, there are three equally likely scenarios, a family of boy-girl, girl-boy, or boy-boy. This gives us a 1 / 3 probability both are boys.