The Two Child Problem: A Probability Puzzle

These puzzles show us that our intuition can mislead us when thinking about probability. If you’d like to share these puzzles with your child, I recommend going through this on your own first to get an understanding because these can be challenging even for adults.

For all questions, assume that each child has an equal chance of being a boy or a girl.

Preparatory Question

Among all families with two children, you pick a family at random. What is the probability that both children in that family are boys?

This question is pretty straightforward, but it is important to do it because it will help make sense of the following ones.

Solution

Make a chart and fill it in with all the possible boy and girl combinations.

Since there is an equal probability of each combination, the probability that both children are boys is 1/4.

Question 1

Among all families with two children in which the older child is a boy, you pick a family at random. What is the probability that both children in that family are boys?

Solution

Use the same chart as you made for the preparatory question, but eliminate all options that are impossible with this new information. Because the older child must be a boy, we can eliminate the last two rows.

Since each possibility has equal odds, there is a 1/2 chance that both children are boys.

This is quite intuitive as well. However, things get more interesting with the next question.

Question 2

Among all families with two children in which at least one of the children is a boy, you pick a family at random. What is the probability that both children in that family are boys?

Solution

Did you spot the difference in the questions? Now you have less information than you did before. You don't know if the older child is a boy or a girl.

Again, let's take the chart from the preparatory question and eliminate all impossible options. Since at least one child must be a boy, the fourth combination from the initial chart is impossible. This leaves us with three possibilities:

The odds that the family has two boys is 1/3.

This question demonstrates that the amount of information we have affects our probability judgements.

Things get even more intriguing in the next and final question.

Question 3

Among all families with two children in which at least one of the children is a boy born on a Tuesday, you select a family at random. What is the probability that both children are boys?

Solution

If you’ve read the solution to question 2, you may be inclined to think that the answer is 1/3 again. However, the probability of both children being boys is now 13/27. Let’s chart out the possibilities.

First, let’s consider the possible older/younger combinations of boys and girls. They are exactly the same as they were in question two:

We now need to consider the days of the week they could have been born.

Write out this chart 3 times – once for each of the three combinations above.
Write “boy” or “girl” on the lines next to “older” and “younger”. Then check off the possibilities of the days of the week they could have been born. Keep in mind that at least one of the children must be a boy born on a Tuesday.

Here are the three completed charts:

To explain the chart above: If the older boy was born on a Tuesday, the younger boy could have been born any day of the week. If the younger boy was born on a Tuesday, the older child could have been born any day of the week.

Since there is only one boy, we know that he must have been born on a Tuesday. The girl could have been born on any day of the week.

Since there is only one boy, we know that he must have been born on a Tuesday. The girl could have been born on any day of the week.

Now that your charts are complete, count up the total number of possibilities—in other words, the total number of checkmarks. There are 27.

Then count the total number of possibilities in which both children are boys. There are 13.

Therefore, the probability of both children being boys in this family is 13/27.

Important Note

The phrasing "Among all families with ___, you pick a family at random" was crucial. These puzzles are not answerable if that is not specified. The Two Child Problem, as it is classically phrased, leaves this out and is therefore too ambiguous to solve. For more about this ambiguity, see the Double-Six Puzzle.