# The Woman With Two Children

Marilyn's critics unconsciously introduce a subtle restraint
into the problem and arrive at a different answer.

A woman has two children, at least one of which is a boy. What is the probability that the other child also is a boy? Marilyn gives the answer as 1/3. Some agree with Marilyn; others insist that the correct answer is 1/2 or declare that the problem is ambiguous.Wiskit

In approaching this problem it is important to consider that the two children are distinguishable. Even if they were identical twins given the same name and it was not determined which of them was born first, we should create a way to distinguish them. The answer to the difference of opinion, we believe, lies in considering whether or not it is known which of the two children is certainly a boy. If it is, then the answer is 1/2; however there is nothing in the problem as stated to suggest that we know this, and so we proceed as follows:

Proceeding from the assumption that boys and girls are equally likely (actually boy births are slightly more common) let us create an ideal population of 100 women. We tell them to have a baby and 50 have a boy and 50 have a girl. We then tell them to have a second baby. Of those who have a boy 25 have another boy and 25 have a girl. Of those who have girls 25 have a boy and 25 have another girl. Our population summary is:

25 B B
25 B G
25 G B
25 G G

We now ask the question "If at least one of your children is a boy (without identifying one) what is the probability that the other child also is a boy?"

We can arrive at an answer by enumerating our ideal population. The number of women "at least one of whose children is a boy" is 75. Of this population, the number of women with two boys is 25, so the probability of the other child being a boy is 25/75 or 1/3.

Some of the people who arrive at the answer 1/2 may do so by mentally simplifying the problem in preparation to solving it. In general that is a useful technique provided that we do not materially change the problem. However in doing so they introduce the subtle assumption that we know which of the two children is certainly a boy. As an example of the usefulness of simplifying a problem consider the following:

A woman has two children at least one of which is not a girl.
What is the probability that at least one of them is not a boy?
Some of the obfuscation may be removed by making equivalent substitutions such as "not a girl" = "boy" and "not a boy" = "girl." The first clause restricts the population to be considered to the same one of 75 as before. The second clause qualifies only the mothers with a girl, of which there are 50, so the probability is 50/75 = 2/3.

Since probability is the science of drawing conclusions based on incomplete information, we cannot complain of too little information to give an answer or of ambiguity based on lack of information. In such cases we must fall back on reasonable presumptions, rebuttable in the face of more information. The presumption of the equal probability of boy and girl births is common in problems of this sort and is generally safe if no birth statistics are given. It corresponds to the state of "zero information" about the frequency of births.

In her book Ask Marilyn, Marilyn reports that this puzzle, referring to two dogs of undetermined sex instead of two children, generated the second largest volume of mail. The Monty Hall Problem, of course, generated the largest. Cory Ondrejka already has contributed to the Monty Hall page. Here is what he has to say about the two children in a message originally sent to Herb:

A small question on the "Marilyn Ignores the Obvious Regarding Probability of Boys" page: how does seeing the boy modify the probability of whether or not she has two sons?

I agree that if you were introduced to her older son that it would change the answer from 1/3 to 1/2, but I don't see how just seeing one son causes this change.

The woman had two children, born either boy girl, girl boy, or boy boy. Those three cases are not distinguishable by observing one of the children, thus the boy boy case is still 1/3 of the total.

Changing it to the older son changes the two children into girl boy or boy boy, thus changing the odds to 1/2.

Likewise, identifying the child as the younger son would have the same result.

Am I missing something obvious?

Cory is raising the question of whether a formal identification is necessary in order to change the probability that the other child is a boy to 1/2. We think so. Probability is a serious topic and probabilities change only with the consideration of additional material information, not with conjecture. Maybe now is the time for the big Indian, little Indian joke:
A big Indian and a little Indian are walking down the street. The little Indian is the big Indian's son, but the big Indian is not the little Indian's father. How is this possible?

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