Boy or Girl Paradox

Here you can learn about the boy or girl paradox: a seemingly innocent question that prompts an intriguing discussion on statistics and probabilities. In this comprehensive article, you can learn the following:

• What is Gardner's boy or girl paradox?: the two questions of the problem.
• The answers to the boy or girl paradox's questions.
• Explaining the boy or girl paradox's ambiguity.

And much more about the importance of how we state questions in science and statistics! Ready?

Martin Gardner, arguably the most important recreational mathematician ever lived, formulated the boy or girl paradox (or the two-child problem) in 1959, in his Mathematical Games column of Scientific American.

The paradox starts with a straightforward set-up: a family with two children of unknown genders. The mathematician then asks two questions and attempts to answer them: one comes with a straightforward answer. The other, surprisingly, has no definitive answer at all!

The following sections will deal with the original boy or girl paradox. To do so, we need to outline the three fundamental assumptions that are often used when dealing with the two-child problem:

• A child can be either a boy or a girl.
• A child has equal probability of being a boy or a girl.
• The gender of a child doesn't depend on the gender of the other (independency).

Let's now see the questions of the boy or girl paradox!

Gardner's boy or girl paradox begins with two simple questions:

1. Mr. and Mrs. Smith have two children. The older one is a boy. What is the chance that the other one is a boy too?
2. Mr. and Mrs. Smith have two children. At least one of them is a boy. What is the chance that the other one is a boy?

Guess or calculate the two answers, then keep reading to find out if you were right or wrong and why you were wrong.

Take the first question of the paradox:

1. Mr. and Mrs. Smith have two children. The older one is a boy. What is the chance that the other one is a boy too?

Here we took the two children, sorted them by age, and identified a situation where the older one is a boy. As there are four possible types of families of two children, we can easily see this situation in a table.

Two of the four combinations can be excluded immediately as the older kid is a girl. This leaves us with two possible situations: boy-boy and girl-boy. According to the assumption made above, the younger kid has an equal probability of being a girl or a boy, independently from the gender of the older one. We then write:

where $P(B_\mathrm{y})$ and $P(G_\mathrm{y})$ are the probabilities of the younger kid being either a boy or a girl.

Solving the simple system of equations answers the two-child problem's first question: $P(B_\mathrm{y})=0.5$, or $1/2$ if you prefer to see the result as a fraction.

The second question of Gardner's boy or girl paradox is:

2. Mr. and Mrs. Smith have two children. At least one of them is a boy. What is the chance that the other one is a boy?

The common sense answer would be $1/3$, right? As there are three possible situations out of the four two-children families where at least one of the kids is a boy (the only one we don't count is the one where both kids are girls), and are all equally likely, it makes sense that the family with two boys happens with probability $1/3$. Well, we are here to show you that this answer is only partially correct, and this depends on how we formulate the question: the one above is purposefully ambiguous!

In one situation, imagine if we asked you to tell us the probability of a family having two boys if you know that one of their children is a boy. In this case, the result would be $1/3$. How do we find it? Let's consider the table we used in the previous answer, but this time ignoring the kids' age.

In this case, we must exclude only the last combination, as it doesn't satisfy the request that one of the children is a boy. This leaves us with three possible family types with at least one boy: each of them has a probability of $1/3$, including the family type with two boys. In this case, we selected the combination of kids after fixing our information regarding the children.

What if, instead, the question "at least" would be formulated starting from a random selection of a kid from all possible family types? Given that this kid is a boy (thus, the family has, at least, a boy), what is the chance of the other kid being a boy?

In this case, the answer would be $1/2$, as for the previous question. However, the mathematics behind this result is different. A good explanation for the boy or girl paradox's second question uses the Bayes theorem. Bayes theorem's formula allows us to compute a conditional probability:

where:

• $P(X|Y)$ — Conditional probability of the event $X$ given the happening of event $Y$. In our case, this quantity is the family's probability of having two boys, given that the first one that you extract at random is a boy, too: $P(BB|B)$. In jargon, this quantity is the posterior probability.

• $P(Y|X)$ — Likelihood, or probability of the event $Y$ given the event $X$ happened with outcome $X$. In our case, this quantity corresponds to the probability of extracting a boy from a family with two boys: $P(B|BB)$.

• $P(Y)$ — Prior probability: in our case, the probability of picking the family type with two boys, $P(BB)$.

• $P(X)$ — Normalization factor that causes the posterior probability to belong in the correct range. Explaining the boy or girl paradox, this quantity corresponds to the probability of extracting a boy from each family.

Let's assign values to these probabilities!

• The likelihood $P(B|BB)$ has a value of $1$ as we will surely extract a boy from the family type with two male children.

• The prior probability $P(BB)$ is easily computed starting from the initial assumptions: as the gender of a kid is uniformly distributed and independent, each family type has equal probability $1/4$ to be the outcome of a random extraction.

• The normalization factor is slightly trickier to compute. $P(B)$ depends on the probability of picking a boy in every possible family type. Hence, we use the following formula:

If you have any doubts about the values we used above, check the following table:

Once we set all the values, we can substitute them in Bayes formula:

The probability of finding a family with two boys, given that at least one of their children is a boy, in this case, is $1/2$!

The ambiguity of the answer to the boy or girl paradox

The second question of the problem is ambiguous: the answer further depends on other information we don't possess at the time. Let's see two possible questions that would give unambiguous answers, one for each of the results we've seen above:

• For the results $1/3$, we can ask ourselves, "Mr. and Mrs. Smith have two children. Given that one of them is a boy, what's the probability of both of them being boys?". In this case, we retained all the possible uncertainty on the family composition: the first boy can be the older or younger one, or this information may not matter (as both children are boys).

• For the result $1/2$, imagine meeting Mrs. Smith with one of her children. In this case, we know that at least one of their children is a boy. What is the probability that the other child is a boy, too? In this case, Mrs. Smith decided to leave the other child at home, regardless of the age or any other information: she didn't pick the older or the younger child; she picked the boy! This leaves us with a probability of $1/2$ for the other kid to be a boy, according to the initial assumptions.

We explain the boy or girl paradox in a fun and interactive way in our tool: follow the instructions, and start making guesses about the results, either by choosing probability from a list or by inserting them: this paradox is a stimulating example of the importance of language in science: you can use it in your classes, for experiments, and much more! You will be surprised by how many people overlook the question's ambiguity. What do you think is the most likely outcome?