You’re friends with someone (let’s call him Mr Smith) who has two children. You know that the older child is a girl. What’s the probability of the younger child also being a girl?
This is one of a set of probability-related problems which are known by various names, such as the ‘two child problem’ or the ‘boy or girl paradox’. This first problem that I’ve outlined is certainly quite a simple one to solve: there is a 50% chance that the younger child is a boy, and a 50% that the younger child is a girl. The answer is therefore 50%, or 1/2.
We can also solve this problem in a more analytical way by examining the sample space, a list of all the possible outcomes. For two children, the sample space consists of four possible outcomes:
older child |
younger child |
boy |
boy |
boy |
girl |
girl |
boy |
girl |
girl |
Because there are four possible outcomes, each which has an equal probability of occurring, we know that each outcome has a 25% chance of occurring.
How does this sample space help us? We can eliminate the outcomes which don’t agree with the information we know and completely ignore them. In this first problem, we know that the older child is a girl, and this eliminates two of the outcomes.
older child |
younger child |
boy |
boy |
boy |
girl |
girl |
boy |
girl |
girl |
Here I’ve coloured blue the outcomes which do agree with the information we know. Now, I will colour green the desired outcomes out of these blue ones (the younger child is also a girl).
older child |
younger child |
boy |
boy |
boy |
girl |
girl |
boy |
girl |
girl |
Because each outcome is equally likely, it’s clear that the probability of both children being girls is 1/2 (the number of green outcomes divided by the number of blue or green outcomes).
This may seem like a convoluted method, but it’s far more useful when we investigate slightly more complicated problems, such as this next one: a second person (let’s call him Mr Jones) also has two children. This time, you only know that at least one of his children is a girl (i.e. the girl could be the older child or the younger child). What’s the probability that both children are girls?
We can construct the sample space again, colouring blue the outcomes which agree with the information we know and colouring green the desired outcome.
older child |
younger child |
boy |
boy |
boy |
girl |
girl |
boy |
girl |
girl |
As you can see, this time there are three outcomes which agree with the information we know, because the girl is not limited to the older child. Here it’s clear that the probability is not 1/2 which may have been expected, but instead 1/3.
The third problem I’ll give is, in my opinion, quite mind-boggling. A third person (let’s call him Mr Williams) also has two children, and you see one of them, a girl, through the window of their house. What’s the probability of both children being girls?
You might expect the answer to be again 1/3, and you could show this by constructing the same sample space as above. However, the mistake in this is that by seeing the girl through the window, you’ve removed the possibility of the child you haven’t seen being the only girl. The sample space would look something like this:
child you saw through the window |
child you didn’t see through the window |
boy |
boy |
boy |
girl |
girl |
boy |
girl |
girl |
By tweaking the sample space a little, we can see that the probability is actually 1/2. In essence this is a repeat of the first problem but with a little twist in how the problem is stated.
This series of problems is quite perplexing in that the answer depends on how you found out the information that one of the two children is a girl. If someone told you that at least one of the children is a girl, then the answer is 1/3. On the other hand, if you physically see or hear one girl, or were told that the older child is a girl, or gained the information in any way which distinguishes between the two children, then the answer is 1/2.
Yanhao