The Two Child Problem states
In a family with two children, what are the chances, if at least one of the
children is a girl, that both children are girls?
It is well attested that the answer is 1 in 3. You can view the other question for an explanation why.
I thought up this solution to the same problem, but since the answer is not the same, there must be some part of my reasoning that is unsound, but I can't quite figure out where the problem is. This is my reasoning:
- If it is given that the older child is a girl, the probability that the younger child is also a girl, and thus that both children are girls, is 1 in 2.
- Similarly, if it is given that the younger child is a girl, the probability that the older child is also a girl, and thus that both children are girls, is also 1 in 2.
- If we know that one of the children is a girl, then it can easily be concluded that either the younger child or the older child is a girl.
- Either way, the probability is still 1 in 2.
Where have I gone wrong?
Best Answer
For a more intuitive reason than the formal answer:
Let's say I use your first two bulleted statements to guess how often both children are female.
When the older child is female, half the time I will guess that both children are female.
Similarly, when the younger child is female, half the time I will guess that both children are female.
Seems consistent, right? However, the logic fails when both children turn out to be girls! In that case, both of your probability statements fire at the same time, and the problem is that I might end up saying "Both children are girls" twice in the same universe. When that happens, we're overcounting the number of times that both children are girls, which is why your answer ended up too large.