The Two Children Problem
You meet a father and son, whose family was selected at random from among all two-child families where the eldest child is a boy. What are the chances that the man’s other child is also a boy?
The next day, you meet a different father and son, whose family was selected at random from among all two-child families with at least one boy. What are the chances that the man’s other child is also a boy?
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|
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
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 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
Son Born on a Tuesday
Now suppose you encounter a mother and son, whose family was selected at random from among all two-child families with at least one boy born on a Tuesday. What is the probability that her other child is a boy?
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
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.