Boys and Girls Problems

The Two Children Problem

You meet a man who says he is the father of two children. “This is my eldest,” he says, indicating the boy with him. What are the chances that his other child is also a boy?

Father and Eldest

The next day, you meet a different man, who also says he is the father of two children. “This is Aaron,” he says, indicating the boy with him. What are the chances that his other child is also a boy?

Father and Aaron

For the purposes of both questions, assume that a child must be either a girl or a boy, that girls and boys exist in equal numbers in the population, and that the gender of a child is independent of the gender of the child’s sibling. If you think you have it figured out, read the answer and explanation below. Then scroll down for more boys and girls problems.

See Answer and Explanation

It may seem like both questions should have the same answer, but that is not the case.

For the first question, the probability that the other child is also a boy is 1/2.

For the second question, the probability that the other child is also a boy is 1/3.

To understand why, begin with the following table, which shows the four gender-distribution possibilities for a two-child family, each of which has 25 percent probability:

  Older Child Younger Child
1 Boy Boy
2 Girl Girl
3 Boy Girl
4 Girl Boy

For the first question, with the information we are given, that the older child is a boy, we have narrowed down the possibilities to #1 or #3, so there is a 1/2 probability that the other child is also a boy.

For the second question, the information we are given, that one of the children is a boy, narrows down the possibilities only to #1, #3, or #4. Among those three choices, in two cases the other child is a girl and in one case the other child is a boy, so the probability that the other child is also a boy is 1/3.

 

Another Boys and Girls Problem

family

Assume that every pregnancy has an equal chance of producing a boy or a girl, and in a certain society, every family follows the policy of having as many children as they can until they have a boy, and then they stop having children. So a family that has a boy first will have only one child, but a family might have one, two, three, or more girls before having their first boy and then having no more children. In this society, after many generations, what will the ratio of boys to girls be?

See Answer
The population will be evenly divided between boys and girls. Each pregnancy has an equal chance of producing a boy or a girl. When a family decides to stop having children does not change that fact.

 

The Four Children Problem

Now suppose you hear of a family with four children, and you do not know the gender of any of them, but you wish to guess the most probable scenario. You reason that four boys or four girls is very unlikely. Three of one gender and one of another seems unlikely too. Since the probability for each child is 1/2, you decide the most likely breakdown is two girls and two boys. Are you correct?

See Answer
Many people are surprised to learn that the scenario of three of one gender and one of another is most likely, as it will occur half the time. Four of one gender will occur in 1/8 of the cases, and the two boys and two girls scenario will occur 3/8 of the time.

 

Son Born on a Tuesday

Now suppose you encounter a woman who says, “I have two children, and one is a son born on a Tuesday.” What is the probability that her other child is a boy?

Son Born On Tuesday

Make the same assumptions as in the first problem, and also assume that births are equally likely on different days of the week.

See Answer
Let us consider the different scenarios. The woman has two children. If the older child is a son born on a Tuesday, then there are 14 possibilities for the younger child: a boy born on any of the 7 days of the week, or a girl born on any of the seven days of the week. These 14 possibilities are represented by the blue column in the chart below. At this point the probability would be 1/2, as we might expect. Then we need to look at what happens if the older child is not a son born on a Tuesday, and therefore it is the younger child that is a son born on a Tuesday. This is represented by the blue row in the chart. In this case, we have the same 14 possibilities for the older child, except one. The possibility of the older child being a boy born on a Tuesday has already been counted, so it cannot be counted again. This gives us 27 possibilities.

Son Born On Tuesday Chart

 

So while we might have been expecting the probability to be 14/28 or 1/2, we cannot count the “Tuesday boy – Tuesday boy” possibility twice. The real probability of two boys is therefore 13/27. This problem was devised by Gary Foshee.

2 thoughts on “Boys and Girls Problems”

  1. Love your site! I disagree with your answer to the first question, though. Why is birth order relevant at all? In your chart, instead of “older child” and “younger child,” why not just “this child” and “that child?” Birth order is an arbitrary variable. You could just as easily have used height, weight, or any other distinguishing trait.

    1. Glad you like the site, and thanks for the comment! I agree that using birth order as the distinguishing trait is somewhat arbitrary, and you could use another trait to achieve the same effect of narrowing the probability that the other child is a boy from 1/3 to 1/2. (Height or weight might be problematic since there are gender differences for those traits.)

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