Readers Question Marilyn's Answer regarding Probability of Boys

Ask Marilyn ® by Marilyn vos Savant is a column in Parade Magazine, published by PARADE, 750 Third Avenue, New York, NY 10017, USA. According to Parade, Marilyn vos Savant is listed in the "Guinness Book of World Records Hall of Fame" for "Highest IQ."

The material from Marilyn's column quoted in this review originally appeared in Parade Magazine May 26, 1996. The same problem was discussed again on December 1, 1996.

Question

Say that a woman and a man (who are unrelated) each has two children. We know that at least one of the woman's children is a boy and that the man's oldest child is a boy. Can you explain why the chances that the woman has two boys do not equal the chances that the man has two boys?

My algebra teacher insists that the probability is greater that the man has two boys, but I think the chances may be the same. What do you think?

-- Michelle Minikel, Brookfield, Wis.

Answer

I agree with your algebra teacher. The woman may have at least one boy in the three following ways: 1) older boy, younger girl; 2) older girl younger boy; 3) older boy, younger boy. But the man's children may be distributed in only two ways: 1) older boy, younger girl; or 2) older boy, younger boy. So the chances are only 1 out of 3 that the woman has two boys, but the chances are 1 out of 2 that the man has two boys.

The readers disagree!

Harry Eaton <haceaton@aplcomm.jhuapl.edu> and Patrick Maupin <patrick.maupin@amd.com> both wrote to point out that Marilyn is assuming that the woman was randomly selected from the collection of all women having two children, at least one of whom is a boy. They believe that this assumption is not necessarily valid.

Consider the collection of all women with two children. Now, consider the following two experiments:

For each woman, pick one of the two children at random. If that child is a boy, what is the probability that the other child is a boy? (Answer: the probability that both children are boys is 1 out of 2.)
Note that to perform this experiment, we need to know in advance the sex of only one child.
For each woman, if either child is a boy, what is the probability that the other child is also a boy? (Answer: the probability that both children are boys is 1 out of 3, as Marilyn explained.)
Note that to perform this experiment, we must know in advance the sex of both children.

Since the original question states only that we know the sex of one child, Harry and Patrick believe that the first experiment described above is the intended question. They believe that this corresponds to the real world question that might be asked by someone who knows one of the two children but not the other. (If we knew the sex of both children, why ask the question in the first place?)

I agree with the algebra teacher

I believe that Harry and Patrick are correct in pointing out that there are two possible interpretations to the question, and that Marilyn's answer was incomplete in that only one of the two possible interpretations was examined. However, I personally prefer the algebra teacher's interpretation.

Many readers are asking for a better explanation than the one Marilyn offered. Perhaps this will clear things up.

Begin with a group of 100 families, each with two children, distributed as follows:

25 families with oldest child a boy and youngest child a boy.
25 families with oldest child a boy and youngest child a girl.
25 families with oldest child a girl and youngest child a boy.
25 families with oldest child a girl and youngest child a girl.

Of this group, there are 50 families in which the oldest child is a boy. Of those 50 families, there are 25 families in which the youngest child is also a boy. In other words, out of the group of families in which the oldest child is a boy, 50% have two boys.

From the same group, there are 75 families in which at least one child is a boy. Of those 75 families, there are 25 families in which the other child is also a boy. In other words, out of the group of families in which at least one child is a boy, only 33% have two boys.

Harry predicted that this column would generate a lot of mail, just as the goat-in-a-game-show column did. Based upon the mail I've received so far, I belive Harry's prediction is correct.

Here's a copy of Harry's Letter to Marilyn.

http://www.wiskit.com/marilyn.boys.html last updated December 2, 1996 by herbw@wiskit.com