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

Mr. Miller has two children. What is the probability that both children are boys?

This question is pretty easy, 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.

Older	Younger
boy	boy
boy	girl
girl	boy
girl	girl

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

Question 1

Mr. Miller has two children. The older one is a boy. What is the probability that both children 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.

Older	Younger
boy	boy
boy	girl

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

Mr. Miller has two children. At least one of them is a boy. What is the probability that both children 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:

Older	Younger
boy	boy
boy	girl
girl	boy

The odds that Mr. Miller has two boys is 1/3.

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

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

Question 3

Mr. Miller has two children. At least one of them is a boy who was born on a Tuesday. 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. After all, the information about the boy being born on a Tuesday seems completely irrelevant.

However, the probability of both children being boys is now 13/27.

I understand that this is tremendously counter-intuitive. 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:

Older	Younger
boy	boy
boy	girl
girl	boy

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.

		Older: _____
		Mo	Tu	We	Th	Fr	Sa	Su
Younger: _____	Mo
	Tu
	We
	Th
	Fr
	Sa
	Su

Here are the three completed charts:

		Older: Boy
		Mo	Tu	We	Th	Fr	Sa	Su
Younger: Boy	Mo		✓
	Tu	✓	✓	✓	✓	✓	✓	✓
	We		✓
	Th		✓
	Fr		✓
	Sa		✓
	Su		✓

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.

		Older: Boy
		Mo	Tu	We	Th	Fr	Sa	Su
Younger: Girl	Mo		✓
	Tu		✓
	We		✓
	Th		✓
	Fr		✓
	Sa		✓
	Su		✓

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.

		Older: Girl
		Mo	Tu	We	Th	Fr	Sa	Su
Younger: Boy	Mo
	Tu	✓	✓	✓	✓	✓	✓	✓
	We
	Th
	Fr
	Sa
	Su

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 is 13/27.