A few months ago Gary Foshee was scheduled to speak at the Gathering for Gardner. He got up and gave a presentation that was all of three sentences. He said:
I have two children. One is a boy born on a Tuesday. What is the probability I have two boys?
This deceptively simple problem quickly made the rounds. The knee-jerk answer is , of course—the gender of one child doesn’t change the probabilities for the second child. People with a little training in probability know that that reasoning isn’t a valid—it is an exercise in conditional probability. The probability of having two boys given that one child is a boy is . Surely, that’s the correct answer, right? How could the day of the week matter?
It turns out, however, that the probability is an unexpected .
[Note: The real correct answer is 0 or 1 since Gary Foshee either has two boys or doesn't have two boys. The actual question should be "If a randomly-selected family has two children and at least one of them is a boy born on Tuesday, what is the probability that they have two boys?" But that doesn't sound nearly so slick. I'll be consistent and stick with his less-than-perfect wording.]
I’d like to pose a question similar to the Tuesday boy problem, then describe how to compute the probabilities for a whole class of problems like these.
I have two children. One is a left-handed boy. What is the probability I have two boys?
You may assume that the probability of being left-handed is 1/10 and that left-handedness is not a genetic trait.
Stop reading here if you want to think about the problem on your own.
Before we answer the left-handed boy problem, lets look at an easier question.
I have two children. One is a boy. What is the probability I have two boys?
We must think of this in terms of conditional probability. We can compute the probability using this formula:
There are four equally likely options for any two-child family: the first child is a boy and the second child is a boy (written ), the first child is a boy and the second child is a girl (written ), , and . These are shown in the chart below with their probabilities.
However, in our case we can ignore the case because we know that at least one child is a boy.
This problem is particularly easy because all four outcomes are equally-likely. The Tuesday boy problem is trickier. The probability of being born on a Tuesday is and the probability of a non-Tuesday birthday is .
Now there are 16 possibilities. The notation means that the first child is a girl born on a non-Tuesday and the second child is a boy born on Tuesday. The probability of this particular occurance is . The probabilities of the others are calculated similarly:
In the chart below we remove the cases in which there is no Tuesday-born boy.
We see that:
So the answer to the question is:
Let us investigate the general case.
I have two children. One is a boy with a trait which occurs with probability . What is the probability that I have two boys?
We build the chart just as we did in the Tuesday boy problem. Here means that the first child is a girl who does not have the trait and the second child is a boy who has the trait.
Again, we focus on those that have a boy with the trait.
Calculating as before we obtain the probability:
Now we can solve a whole class of these problem. To solve the left-handed boy problem we simply plug in to obtain an answer of . That is, if a family has two children and one of the children is a left-handed boy, then the probability that they have two boys is .
If we had asked the “right-handed boy” problem, then we’d plug in to obtain . That is, if a family has two children and one of the children is a right-handed boy, then the probability that they have two boys is .
Notice what happens in the limiting values of .
If the trait is very common, like “has two eyes,” then and we’re essentially in the “I have two children and one of them is a boy” case. Accordingly we have .
On the other hand, if we have an extremely rare trait, like “has climbed Mt. Everest” (), then it is very unlikely that both children have this trait. We’ve essentially uniquely identified one of the children. If we looked at all the two-child families in the entire world that have a son who climbed Mt. Everest, very few of them will have another child who also climbed Mt. Everest. Most of them are a boy who climbed Mt. Everest, and one other child. The chance of the other child being a boy is 1/2. (It is like asking “My first born child is a boy. What is the probability that I have two boys?”) Accordingly, .
I’d like to thank my colleagues Jeff Forrester and Barry Tesman for their helpful comments.