The Boy or Girl Paradox

• If a family has two children and the older one is a boy, what is the probability that the younger child is a girl?
• If a family has two children and at least one of them is a boy, what is the probability that the family also contains a girl?

These questions both look quite innocent, and most people will quickly reply that since the probability of a child being a girl is ½, the answer to both the above questions is ½. We could easily leap to this conclusion too, but we would, of course, be wrong. Though they may seem to be asking the same thing, these questions actually lead to different answers and are the basis of a mildly confusing problem known as the Boy or Girl Paradox.

The Correct Answers

To be fair, ½ is in fact the correct answer to the first question. Let us represent a boy with 'B' and a girl with 'G', with the older child coming first, and assume that the boy:girl ratio is precisely 50:50. This produces four possible combinations: BB, BG, GB and GG. However, in the case of the first question, we are told that the older child is a boy, thus rendering the combinations GB and GG impossible. We are thus left with BB and BG, in which one out of the two equal possibilities contains a girl as the younger child. The probability of the younger child being a girl is thus ½.

• Two children → combinations are BB, BG, GB, GG.
• The older child is a boy → combinations are BB, BG.
• Probability of one child being a girl is thus ½.

Now let us look at the second question, which states that at least one of the children is a boy. This means that out of the four possibilities, only GG is impossible owing to the fact that it does not contain a boy. As the second question does not state whether the boy is the older or the younger child, it is possible to have any one of GB, BG or BB. In other words, the boy we know of could have an older sister, a younger sister or a brother¹.

Note that the last possibility, BB, should only be counted once. This point can be confusing and thus merits a further explaination. First, let us look at GB and BG:

• GB = there is a younger boy who has an older sister.
• BG = there is an older boy who has a younger sister.

Clearly, these two situations are different, and thus represent two distinct possibilities. However, let us treat the ways in which BB might occur in the same manner:

• BB = there is a younger boy who has an older brother.
• BB = there is an older boy who has a younger brother.

Unlike the first pair of sentences, the ones for BB both describe the same situation - the words we use to describe BB simply depends on which of the boys we think the question has already referred to. BB is therefore only one possibility out of three, and thus has a ¹⁄₃ probability of occurring. On the other hand, having an older boy and a younger girl is different to having an older girl and a younger boy, and the probability of the family including a girl is therefore ²⁄₃.

• Two children → combinations are BB, BG, GB, GG.
• At least one child is a boy → combinations are BB, BG, GB.
• Probability of one child being a girl is ²⁄₃.

Why Is It A Paradox?

The above puzzle is referred to as a paradox simply because the solution is quite counter-intuitive. Having looked at the first question and decided on an answer of ½, the average person will decide that there is no real difference between the two questions, thus reasoning that the answer to both questions is exactly the same. They will then become rather upset upon being told that they are wrong and will argue that the person showing them the puzzle is being very silly, and walk off in a huff. This last step is part of many paradoxes, and also occurs as a result of trying to explain the Monty Hall problem, which is in fact even more confusing than this one.

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¹ It's also possible that the two children are twins, but if we assume that one was born just before the other we can avoid complicating the situation.