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.

Observe that:
, and
.
So,
.
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.