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The Boy or Girl paradox is a well-known set of questions in probability theory, dating back to at least 1979^[1] and 1959. There are many variants. Martin Gardner published one of the earliest variants of the paradox, published in Scientific American in 1959 described the problem as The Two Children Problem:

• Mr. Jones has two children. The older child is a girl. What is the probability that both children are girls?"
• Mr. Smith has two children. At least one of them is a boy. What is the probability that both children are boys?

Other variants of this question have been recently popularized by Ask Marilyn in Parade Magazine, John Tierney of The New York Times^[2], Leonard Mlodinow in Drunkard's Walk.^[3], as well as numerous online publications^[4][5][6][7] The paradox stems from the intuition that the answer to these questions is the same^[3][5][8]. Scientific studies have shown that over 88% of MBA studentsstudied answer a unambiguously-worded version of the second question incorrectly.^[8] This paradox has stimulated a great deal of contraversy, largely in response to the variant posed by Marilyn Vos Savant^[9]. Many people, including professors of mathematics, argued strongly for both sides with a great deal of confidence^[9], sometimes showing disdain for those who took the opposing view.^[9] Subsequent inquires ^[8][10][9] have shown that although the answer depends on the wording, carefully phrased versions of the paradox provide different answers to the two questions.

There are four possible combinations of children. Labeling boys B and girls G, and using the first letter to represent the older child, the possible combinations are:

{BB, BG, GB, GG}.

These four possibilities are taken to be equally likely a priori. This follows from three assumptions:

1. That the determination of the sex of each child is an independent event.
2. That each child is either male or female.
3. That each child has the same chance of being male as of being female.

It is worth noting that these conditions form an incomplete model. By following these rules, we ignore the possibilities that a child is intersex, the ratio of boys to girls is not exactly 50:50, and (amongst other factors) the possibility of identical twins means that sex determination is not entirely independent. However, one can see intuitively that the occurrence of each of these exceptions is sufficiently rare to have little effect on our simple analysis of the general population.

• A (random) family has two children, and the older child is a boy. What is the probability that the younger child is a girl?

In this problem, a random family is selected. In this sample space, there are four equally probable events:

Only two of these possible events meets the criteria specified in the question (e.g., BG, BB). Since both of the two possibilities in the new sample space {BG, BB} are equally likely, and only one of the two, BG, includes a girl, the probability that the younger child is a girl is 1/2.

• A (random) family has two children, and one of the two children is a boy. What is the probability that the younger child is a girl?

This question is identical to question one, except instead of specifying that the older child is a boy, it is specified that one of them is a boy. If it is assumed that that information was obtained by considering both children^[11], then there are four equally probable events for a two-child family as seen in the sample space above. Three of these families meet the necessary and sufficient condition of having at least one boy. The set of possibilities (possible combinations of children that meet the given criteria) is:

Thus, the answer to question 2 is 1/3.

However, if it is assumed that the information was obtained by considering only one child, then the problem is identical to question one, and the answer is 1/2.^[12][11]

• A (random) family has two children, and one of the two children is a boy named Jacob. What is the probability that the other child is a girl?

Or, the set {GJ, JG, JB, BJ}, in which two out of the four possibilities includes a girl.

Therefore we might think that the probability returns to 1/2. But this is wrong because it doesn't take into account different frequencies of each of these answers. The likelihood of a boy being named Jacob and a boy not being named Jacob are not equal. Thus, we must replace our classical interpretation of probability with either a Frequentist or Bayesian interpretation. (Note that in real life child names are not independent of each other. In particular, people usually do not give the same name to two children. Thus, this discussion is purely theoretical).

Consider 10,000 families that have two children. Assume that the gender and name of each child is independent, within family and between families. Assume that the probability of each individual child being a girl is .5; otherwise the child is a boy. Assume that the probability of a child having the name Jacob is .01, and that all children with name Jacob are also boys.

In the table above, we have a list of all possible unique outcomes. But these outcomes do not have the same frequency. If we start with the assumption that the family has two children, we get the following frequency table:

With the additional bit of information that the family has a boy named Jacob, we can break every instance of "Boy" into two: "Jacob" and "Boy not Jacob". For every 50 Boys, 1 will fall into the "Jacob" bin and 49 into the "Boy not Jacob" bin. Thus, we have the following table:

If we eliminate all instances that do not meet our given criteria ({Girl, Girl} {Girl, Boy not Jacob} {Boy not Jacob, Girl} {Boy not Jacob, Boy not Jacob}), then we eliminate 9801 of our events, leaving 199 possible events. Of those, the successful events are {Girl, Jacob} and {Jacob, Girl}, or 100 cases.

So if the probability of a boy being named Jacob is 1 in 50, then the probability that the family has a girl is 100/199, or roughly 50%. But this value will change depending on the popularity of the name. At the extreme, if all boys were given the same name, then being named Jacob would provide no more information than being a boy, and thus the probability would still be 2/3 that the family has a girl. As the likelihood of the name decreases, the likelihood of the two-Jacob case also decreases, and the probability of the family having a girl approaches the limit of 50%.

If we further assume that parents never name two children with the same name, we can eliminate {Jacob, Jacob}, leaving 198 possible events; thus it would appear that the probability of the family having a girl is 100/198, or 50/99. However, there are now 50 occurrences each of {Jacob, Boy not Jacob} and {Boy not Jacob, Jacob} making the probability of a girl 100/200, or exactly 1/2.

The second question is often posed in a way that leave multiple interpretations open. In response to reader criticism in response to the question posed in 1959, Gardner agreed that a precise formulation of the question is critical to getting different answers for question 1 and 2^[13]. Specifically, Gardner argued that a "failure to specify the randomizing procedure" could lead readers to interpret the question in two distinct ways^[13]:

• From all families with two children, at least one of whom is a boy, a family is chosen at random. This would yield the answer of 1/3.
• From all families with two children, one child is selected at random, and the gender of that child is specified. This would yield an answer of 1/2.

Whether the question is ambiguous is contraversial. For instance, a subsequent inquiry by Bar-Hillel and Falk argued that although Gardner's original formulation was not ambiguous, slight variations in wording could lead to very different answers^[12] On the other hand, Grinstead and Snell argue that the question is ambiguous in much the same way Gardner did.^[11].

Two old classmates, Mary and Brian, meet in the street, not having seen each other since they left school.

Mary asks Brian: "Have you got any children?"

Brian answers: "Yes, I've got two."

Mary: "Do you have a boy?"

Brian: "Yes, I do!"

Here, for some reason, the conversation is cut short. In this case, Brian does know the genders of both children but was asked specifically about boys, and only if there was one boy among the two children. Mary only knows that "at least one is a boy" because she asked for incomplete information. Accordingly, the probability that Brian has a girl is 2/3.

Mr. Smith is the father of two. We meet him walking along the street with a young boy whom he proudly introduces as his son. What is the probability that Mr. Smith’s other child is also a boy?

Again, all we know about Mr. Smith's children is that "at least one is boy." But since we didn't actually met the other child, that child's gender cannot be influenced by this knowldege. It does not matter whether the child we met is the elder or the younger, only that we met just one child. Bar-Hillel and Falk show that the probability that Mr. Smith's other child is a boy is indeed 1/2.

The Boy or Girl paradox is of interest to psychological researchers who seek to understand how humans estimate probability. For instance, Fox & Levav (2004) used the problem (called the Mr. Smith problem, credited to Gardner, but not worded exactly the same as Gardner's self-admitted ambiguous version) to test theories of how people estimate conditional probabilities.^[8]. In this study, the paradox was posed to participants in two ways:

• “Mr. Smith says: ‘I have two children and at least one of them is a boy.’ Given this information, what is the probability that the other child is a boy?”
• “Mr. Smith says: ‘I have two children and it is not the case that they are both girls.’ Given this information, what is the probability that both children are boys?”

The authors argue that the first formulation gives the reader the mistaken impression that there are two possible outcomes for the "other child"^[8], whereas the second formulation gives the reader the impression that there are four possible outcomes, of which one has been rejected. The study found that 88% of participants provided the incorrect answer (e.g., 1/4 or 1/2) for the first formulation, whereas only 63% of participants responded incorrectly on the second formulation. The authors argue that the reason people respond incorrectly to this question (along with other similar problems, such as the Monty Hall Problem and the Bertrand's box paradox) is because of the use of naive heuristics that fail to properly define the number of possible outcomes^[8].

• Three cards problem
• Monty Hall problem
• Marilyn vos Savant

• Martin Gardner. The Second Scientific American Book of Mathematical Puzzles and Diversions. Simon & Schuster, 1961. Republished by University Of Chicago Press, 1987, ISBN 978-0226282534.

• Boy or Girl: Two Interpretations
• A Problem With Two Bear Cubs
• Lewis Carroll's Pillow Problem