Last year (wow…time flies), I posted a solution to the Two Child problem using Bayes theorem. If you are unfamiliar with this problem, you may want to read that post first.
There has continued to be discussion on this topic on the LinkedIn group where I was originally introduced to it. One of the comments to my previous post summarizes many of the issues that were brought up.
Let me summarize:
One of the objections is that you have to consider why the person is providing this information. Knowing something about why you were provided the information you have and why you were not provided information you don’t have can certainly be used to improve your inferences.
However, without this additional information, you have two choices: (1) do nothing but worry about what would happen if you did have the additional information, or (2) proceed to make an inference based on the information you *do* have.
Many times when solving a problem, I wish that I had more information. But then I am reminded of the words of my grandmother, “Wish in one hand and spit in the other, and see which gets full first!”
When solving an inference problem, you must use only the information at hand—not the information you want (sounds like a Donald Rumsfeld quote). In the current problem, you are simply informed that “one of the children is a boy”. You do not know which child the person is referring to, and you do not know why you were provided with that information.
As described above, there are two ways to proceed with this problem. Solve the problem at hand, which is what I illustrated previously using Bayes Theorem. Or expand the problem by considering why you were provided the information that you have, rather than some other information. If the information came to you by means of a person informing you, you might consider the intention or motivation of that person. If you know something about that person already, this knowledge could be helpful. However, this changes the problem.
Keep in mind now that this is a puzzle. And one (of the many) purpose(s) of a puzzle is to educate. By changing the problem, you lose the opportunity to learn something. So let’s focus on the puzzle that was posed and resist temptation to consider variants. In my own personal experience, I find such temptations to indicate the fact that I don’t know how to solve the puzzle that was presented to me.
What I showed in the previous post was that if you know that a person has “two children at least one of whom is a boy” then the probability is 1/3 that both are boys; whereas if you know that a person has “two children one of whom is a boy born on a Tuesday” the probability that both are boys changes to 13/27.
What we learn is that *any* information about the child improves your inference as to whether both children are boys. I found this to be quite shocking, especially since the information that the child was born on Tuesday is at first glance irrelevant.
In the solution using Bayes theorem, you can see that this information about the day the child was born *does* indeed affect your inferences by coming in via application of the sum rule where you subtract off a term that involves the possibility that both children were boys born on Tuesday. This makes sure that you do not double-count that particular case.
In fact, the more unlikely that case (both being boys born on Tuesday), the more distinguishable the children are from one another.
One can now easily see that *any* information that enables you to distinguish one child from the other will improve your inference. Let’s take a closer look at the problem from this perspective by considering several different states of knowledge and the probabilities that a rational agent would infer that both children are boys. I will denote this probability by Prob(BB) and leave proofs to the reader.
“I have two children” Prob(BB) = 1/4
“I have two children at least one of whom is a boy” Prob(BB) = 1/3
“I have two children at least one of whom is a boy born on a Tuesday” Prob(BB) = 13/27
“I have two children, and here in front of you is my son Bob” Prob(BB) = 1/2
In the least informed case, you only know that the person has two children, so the probability that you would expect both to be boys is 1/4. In the most informed case, where you meet one of the children, the probability that both are boys is equal to the probability that the other un-met child is a boy (1/2). As you start from the information that there are two children and learn more and more about the children, the probability that both are boys changes from a minimum of 1/4 to a maximum of 1/2.
Focusing on the puzzle rather than being distracted by all the other puzzles you could have been solving enables you to learn something from it. Here we find that this puzzle is about the problem of distinguishability versus indistinguishability. And we see that it is not at all trivial.