Boy or girl paradox

The Boy or Girl problem is a well-known example in probability theory:

• In a two-child family, the older child is a boy. What is the probability that the younger child is a girl?
• A two-child family has at least one boy. What is the probability that it has a girl?

There are variations in the exact wording; often the second question confusingly asks about the "other child".

Although these questions appear to be equal at first glance, at deeper investigation they turn out to be very different and to lead to different answers as well.

This problem is so small that it can be solved by exhaustively looking at all combinations in the sample space.

Common assumptions

There are four possible combinations of children. Labeling boys B and girls G, and using the first letter to represent the older child, the sample space is

{BB, BG, GB, GG}.

These four possibilities are taken to be equally likely a priori. This prior follows from two assumptions: that the determination of the sex of each child is an independent event, and that each child has the same chance of being male as of being female. It is helpful to acknowledge these assumptions, not least because in the real world neither is true. The ratio of boys to girls is not exactly 50:50, and (amongst other factors) the possibility of identical twins means that sex determination is not entirely independent.

First question

• In a two-child family, the older child is a boy. What is the probability that the younger child is a girl?

When the older child is a boy, then the elements {GG} and {GB} of original sample space cannot be true, and must be deleted so that the problem reduces to:

{BB, BG}.

Since only one of the two possibilities in the new sample space, {BG}, includes a girl, the probability that the younger child is a girl is 1/2.

Second question

• A two-child family has at least one boy. What is the probability that it has a girl?

In this question the order or age is not important. Therefore the set is:

{BG, GB, BB}

Therefore the probability is 2/3.

Conclusion

The majority of the people coming across this paradox for the first time will agree with the answer to the first question, but will consider the second as nonsense as it has to be of course the same as the first.

This erring can be explained in many different ways.

The main reason is that the second question does not assume anything about the age of the boy, he might be the older and he might be the younger sibling. Therefore the loose thought that there are only 3 possibilities (2 boys {BB}, 2 girls {GG} or a mix) does not take into account that the latter is twice as likely than the formers, because it can be either {GB} or {BG}.

Another way of explaining is: the chance that there are 2 boys is 1/4, same as the chance that there are 2 girls. The chance that there is one boy and one girl (or one girl and one boy) consumes the remainder (1/2), therefore two boys are half as likely as a mixture.

Mistakes

A look why some "explanations" are flawed can also be very explanatory.

For example to answer the second question someone may make this list of possibilities:

1. The boy has an elder brother
2. The boy has a younger brother
3. The boy has an elder sister
4. The boy has a younger sister

Apparently only the latter two are the ones sought for, giving a total probability of 1/2. The error here is that the first two statements are counted double. We do not know which brother is the older, as that was not stated in the question. Call the brothers Tom and Harry.

1. Harry has an elder brother Tom
2. Tom has a younger brother Harry

The second statement repeats the first and therefore should be removed.

An ambiguous real-life version

Two old classmates, Anna and Brian, meet in the street, not having seen each others since they left school.

1. Anna asks Brian: "Have you got any children?"
2. Brian answers: "Yes, I've got two."
3. Anna: "Do you have a boy?"
4. Brian: "Yes, I do!"

Here, for some reason, the conversation is cut short.

Formally, this corresponds to the second version as Brian only has told Anna that at least one child is a boy. Accordingly, the probability that Brian has a girl should be 2/3. However, in real conversation, if Brian had two boys, he would be more likely to answer e.g. "Yes, they are both boys!" The fact he does not answer like that could reasonably be taken by Anna as a clue increasing her posterior probability of one child being a girl above 2/3. This highlights the need for precision when stating such problems in probability.

See also

• Three cards problem
• Monty Hall problem

External links

• http://mathforum.org/library/drmath/view/52186.html
• A Problem With Two Bear Cubs
• Lewis Carroll's Pillow Problem