Boy or Girl paradox

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

  • A random two-child family whose older child is a boy is chosen. What is the probability that the younger child is a girl?
  • A random two-child family with at least one boy is chosen. What is the probability that it has a girl? (Or: in a random two-child family, one of the children is a boy. What is the probability that the other one is a girl?)

Investigation of these questions reveals that their answers are very different.

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[edit] 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 possible combinations are:

{BB, BG, GB, GG}.

These four possibilities are taken to be equally likely a priori. This follows from three assumptions:

  1. That the determination of the sex of each child is an independent event.
  2. That each child is either male or female.
  3. That each child has the same chance of being male as of being female.

It is worth noting that these conditions form an incomplete model. By following these rules, we ignore the possibilities that a child is intersex, 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. However, one can see intuitively that the occurrence of each of these exceptions is sufficiently rare to have little effect on our simple analysis of the general population.

[edit] First question

  • A random two-child family whose older child is a boy is chosen. 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:

Older child Younger child
Girl Girl
Girl Boy
Boy Girl
Boy Boy

Or, the set {BG, BB}.

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

[edit] Second question

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

An equivalent and perhaps clearer way of stating the problem is "Excluding the case of two girls, what is the probability that two random children are of different gender?"

Neither order nor age is important. There are four possible child combinations for a two-child family as seen in the sample space above. Three of these families meet the criteria of having at least one boy. The set of possibilities (possible combinations of children that meet the given criteria) is:

Older child Younger child
Girl Girl
Girl Boy
Boy Girl
Boy Boy

Or, the set {GB, BG, BB}, in which two out of the three possibilities includes a girl.

Therefore the probability is 2/3.

[edit] Bayesian approach

Consider the sample space of 2-child families.

  • Let X be the event that the family has one boy and one girl.
  • Let Y be the event that the family has at least one boy.
  • Then:
    • P( X | Y ) = \frac{ P( X \cap Y ) } { P( Y ) } = \frac{ \frac{2} {4} } { \frac{3} {4} } = \frac{2}{3}

[edit] Conclusion

Many people coming across this paradox for the first time will agree with the answer to the first question, but some may be confused by the answer to the second question.

Two ways of explaining the error are as follows:

  1. The second question does not assume anything about the age of the boy. He might be the older or he might be the younger sibling. Therefore the thought that there are only three possibilities (2 boys {BB}, 2 girls {GG}, or a mix) does not take into account that the last of these three is twice as likely as either of the first two, because it can be either {GB} or {BG}.
  2. The chance that there are two boys is 1/4, the same as the chance that there are two 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.

[edit] Mistakes

A look why some "explanations" are flawed can 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. If there are two boys, we have no referent for "the boy". Therefore the first two possibilities should read:

  1. A boy has an elder brother
  2. A boy has a younger brother

But now it is clear that these two statements are equivalent – both effectively state that there are two boys – and therefore one should be removed.

[edit] An ambiguous real-life version

Two old classmates, Anna and Brian, meet in the street, not having seen each other 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," (Grice's maxim of quantity). The fact that 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.

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