*by Douglas Zare*

9 January 2015

Douglas Zare

Backgammon players have a lot of practice working with objective probabilities and risks. These are things many people find counterintuitive. In many cases, backgammon players may be better able to avoid probability fallacies and errors. Let's look at a probability puzzle where many people have trouble, but where backgammon players might have an advantage.

Martin Gardner wrote about a puzzle called the *Two Children Problem*.

- Suppose a family has two children. The older child is a boy. What is the probability that the younger is a boy?
- Suppose a family has two children. There is at least one boy. What is the probability that both are boys?

We should assume that half of children are boys, and ignore twins and other correlations between the sexes of siblings. These are not exactly right, but these complexities are unimportant and not the point. The sex of each future child may be viewed as an independent coin flip.

The answer in the first situation is 1/2. Knowing the sex of the first child does not tell you anything about the sex of the second. This should surprise no one.

The second situation is not obvious. The answer is 1/3, and many find that counterintuitive. Here is one approach to see the answer is 1/3. If there are two children, there are 4 equally likely possibilities: BB, BG, GB, and GG, where we label the children in order of age. Knowing that there is at least one boy eliminates GG, but accepts all cases of the other 3 possibilities. Only in the case BB are both children boys, and that is 1 out of 3.

Many people expect the answer to be 1/2. People are surprisingly bad at handling probability, but this is usually because we use ideas that work elsewhere out of context. It is rare to encounter precisely the information that there is at least one boy without anything else (see below), and people are not used to interpreting a statement like that. Many true statements come attached to subtle additional information. One of the times you get a statement like "at least one is a boy" with nothing extra is in backgammon!

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