# Discussion of "The Other Child" problem

## The Other Child

A man says, "I have two children; at least one of them is a boy." What is the probability that the other one is a boy?

## Solution

### Incorrect solution

Assuming that girls are born with probability one half, for the families with two children, there are four equally likely cases:

1. 1st child - boy, 2nd child - boy;
2. 1st child - boy, 2nd child - girl;
3. 1st child - girl, 2nd child - boy;
4. 1st child - girl, 2nd child - girl.

When the man states that he has at least one boy, this eliminates the last case, leaving us with three cases. In only one of these cases the other child is also a boy, which makes the probability one third.

### Another incorrect solution

There are twice as many families with a boy and a girl, than with two boys. For the families with a boy and a girl, the father would either say that he has at least one boy or that he has at least one girl. This means that the conditional probability that he would say that he has at least one boy while having a boy and a girl is one half. Because there exist twice as many families with a boy and a girl than with two boys, the father would say that he has at least one boy for half of the cases with a boy and a girl and for all of the cases with two boys. The fact that he said that he has at least one boy makes it equally probable for the other child to be a boy or a girl. This means that the answer is 1/2.

### Correct solution

We must assume that the man is telling the truth. To solve this problem we need to know the probabilities of what the man says depending on the type of family he has. Suppose that if he has two boys, then he would say that he has at least one boy with the probability a. Also, suppose that if he has a boy and a girl then he would say that he has at least one boy with the probability b. Then the probability that he has two boys is a/(a+b). There can be several different assumptions:

• The case is symmetric towards boys and girls and the man always constructs an answer of the type: "I have at least one boy/girl". In other words, when he has a boy and a girl, with probability one half, he says that at least one child is a boy, and with probability one half, he says that at least one child is a girl. When he has two boys, he says that he has at least one boy with probability one. In this case the answer is 1/2.
• We are interested in discussing boys, and a man who has at least one boy always says that he has at least one boy. In this case the answer is 1/3.
• The man wants to have boys and boasts about them as much as possible. In the case that he has one boy and one girl, he says that he has at least one boy with probability one. If he has two boys, he always says that he has two boys. In this case the answer is 0.
• We may invent any distribution of probabilities here.

The correct answer is that the problem is undefined. For example, there is no indication in the problem that all fathers with at least one boy will tell you, "I have at least one boy."

### Discussion continues

Usually, problems of this kind are of two types: either families with at least one boy are equally distributed (in this case the probability is 1/3), or boys in the families with at least one boy are equally distributed (in this case the probability is 1/2). In my experience, all problems where families are equally distributed are formulated artificially. It is difficult to think of such problems. In almost all real life situations boys are the ones that are equally distributed.

Last revised April 2006