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Boy or Girl?

In a two-child family, one child is a boy. What's the probability that the other child is a girl?

While our intuition may tell us that the probability the other child is a girl should be 1/2, or 50%, mathematicians argue that the correct probability could be 2/3, or 67% (assuming that the probability of having a boy or a girl is equally likely for each birth.) Why?



Choosing the Family First

Using one interpretation of the problem, we randomly choose a two-child family. Once the family has been selected, we determine that at least one child is a boy. (For example, from all the mothers with two children, we select one and ask her whether she has at least one son.) In this case, an unambiguous statement of the question could be:

From the set of all families with two children, a family is selected at random and is found to have a boy. What is the probability that the other child of the family is a girl?

Let's look at the possible combinations of two children. We'll use B for Boy and G for girl, and for each combination we'll list the older child first. There are four possible combinations:

BG and GB are listed separately because BG represents a family in which the oldest child is a boy, while GB represents a family in which the oldest child is a girl. Each of the above combinations is equally likely. Since we're told that one child (we don't know which) is a boy, we can eliminate the GG combination. Thus, our remaining possible combinations are:

Each of these combinations is still equally likely.

Now we want to count the combinations in which the "other" child is a girl. There are two such combinations: BG and GB.

Since there are three combinations of possible families, and in two of them one child is a girl, the probability is 2/3.



Choosing the Child First

Supposing, on the other hand, that we randomly pick a child from a two-child family. We see that he is a boy, and want to find out whether his sibling is a brother or a sister. (For example, from all the children of two-child families, we select one at random who happens to be a boy, and ask how many children are in his family and he responds "two.") In this case, an unambiguous statement of the question could be:

From the set of all families with two children, a child is selected at random and is found to be a boy. What is the probability that the other child of the family is a girl?

From the possible combinations of all two-child families we can again eliminate the GG combination, since we know that one child is a boy. As before, the three remaining possible combinations are:

In these combinations there are four boys, of whom we chose one. Let's identify them from left to right as B1, B2, B3 and B4. Of these four boys, only B3 and B4 have a sister, so our chance of randomly picking one of them is 2 in 4, and the probability is 1/2.



Why are these two probabilities different?

As in other probability problems, how information is obtained is as important as the information itself. Without knowledge of the data gathering process, ambiguity can result. How do we know that one child is a boy?

In the first interpretation, each family has an equal chance of being chosen. In a family with two boys, each boy has only half that chance of being "the boy" referenced in the statement. The families are equally probable, but the boys are not.

In the second interpretation, each boy has an equal chance of being chosen. Thus, the family with two boys has twice the chance of being chosen. The boys are equally probable, but the families are not.



For more about this problem, see:

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