The boy or girl paradox surrounds a set of questions in probability theory, which are also known as the two children problem, Mr. Smith's children and the Mrs. Smith problem. The initial formulation of the question dates back to at least 1959, when Martin Gardner featured it in his May 1959 "Mathematical Games column" in Scientific American. He phrased the paradox as follows: 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?
Gardner initially gave the answers and , respectively, but later acknowledged that the second question was ambiguous. Its answer could be , depending on the procedure by which the information "at least one of them is a boy" was obtained. The ambiguity, depending on the exact wording and possible assumptions, was confirmed by Maya Bar-Hillel and Ruma Falk, and Raymond S. Nickerson.
Other variants of this question, with varying degrees of ambiguity, have been popularized by Ask Marilyn in Parade Magazine, John Tierney of The New York Times, and Leonard Mlodinow in The Drunkard's Walk. One scientific study showed that when identical information was conveyed, but with different partially ambiguous wordings that emphasized different points, the percentage of MBA students who answered changed from 85% to 39%.
The paradox has stimulated a great deal of controversy. The paradox stems from whether the problem setup is similar for the two questions. The intuitive answer is . This answer is intuitive if the question leads the reader to believe that there are two equally likely possibilities for the sex of the second child (i.e., boy and girl), and that the probability of these outcomes is absolute, not conditional.
Common assumptions First, it is assumed that the space of all possible events can be easily enumerated, providing an extensional definition of outcomes: {BB, BG, GB, GG}. This notation indicates that there are four possible combinations of children, labeling boys B and girls G, and using the first letter to represent the older child. Second, it is assumed that these outcomes are equally probable. This implies the following model, a Bernoulli process with p = :
Each child is either male or female. Each child has the same chance of being male as of being female. The sex of each child is independent of the sex of the other.
First question Mr. Jones has two children. The older child is a girl. What is the probability that both children are girls?
Under the aforementioned assumptions, in this problem, a random family is selected. In this sample space, there are four equally probable events:
Older child Younger childGirlGirlGirlBoy Boy Girl Boy Boy
Only two of these possible events meet the criteria specified in the question (i.e., GG, GB). Since both of the two possibilities in the new sample space {GG, GB} are equally likely, and only one of the two, GG, includes two girls, the probability that the younger child is also a girl is .
Second question Mr. Smith has two children. At least one of them is a boy. What is the probability that both children are boys?
This question is identical to question one, except that instead of specifying that the older child is a girl, it is specified that at least one of them is a boy. In response to reader criticism of the question posed in 1959, Gardner said that no answer is possible without information that was not provided. Specifically, that two different procedures for determining that "at least one is a boy" could lead to the exact same wording of the problem. But they lead to different correct answers:
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 . From all families with two children, one child is selected at random, and the sex of that child is specified to be a boy. This would yield an answer of .
Grinstead and Snell argue that the question is ambiguous in much the same way Gardner did. They leave it to the reader to decide whether the procedure, that yields 1/3 as the answer, is reasonable for the problem as stated above. The formulation of the question they were considering specifically is the following: Consider a family with two children. Given that one of the children is a boy, what is the probability that both children are boys? In this formulation the ambiguity is most obviously present, because it is not clear whether we are allowed to assume that a specific child is a boy, leaving the other child uncertain, or whether it should be interpreted in the same way as "at least one boy". This ambiguity leaves multiple possibilities that are not equivalent and leaves the necessity to make assumptions about how the information was obtained, as Bar-Hillel and Falk argue, where different assumptions can lead to different outcomes (because the problem statement was not well enough defined to allow a single straightforward interpretation and answer).
For example, say an observer sees Mr. Smith on a walk with just one of his children. If he has two boys then that child must be a boy. But if he has a boy and a girl, that child could have been a girl. So seeing him with a boy eliminates not only the combinations where he has two girls, but also the combinations where he has a son and a daughter and chooses the daughter to walk with.
So, while it is certainly true that every possible Mr. Smith has at least one boy (i.e., the condition is necessary), it cannot be assumed that every Mr. Smith with at least one boy is intended. That is, the problem statement does not say that having a boy is a sufficient condition for Mr. Smith to be identified as having a boy this way.
Commenting on Gardner's version of the problem, Bar-Hillel and Falk note that "Mr. Smith, unlike the reader, is presumably aware of the sex of both of his children when making this statement", i.e. that "I have two children and at least one of them is a boy." It must be further assumed that Mr. Smith would always report this fact if it were true, and either remain silent or say he has at least one daughter, for the correct answer to be as Gardner apparently originally intended. But under that assumption, if he remains silent or says he has a daughter, there is a 100% probability he has two daughters.
Analysis of the ambiguity If it is assumed that this information was obtained by looking at both children to see if there is at least one boy, the condition is both necessary and sufficient. Three of the four equally probable events for a two-child family in the sample space above meet the condition, as in this table:
Older child Younger childGirlGirlGirlBoyBoyGirlBoyBoy
Thus, if it is assumed that both children were considered while looking for a boy, the answer to question 2 is . However, if the family was first selected and then a random, true statement was made about the sex of one child in that family, whether or not both were considered, the correct way to calculate the conditional probability is not to count all of the cases that include a child with that sex. Instead, one must consider only the probabilities where the statement will be made in each case. So, if ALOB represents the event where the statement is "at least one boy", and ALOG represents the event where the statement is "at least one girl", then this table describes the sample space:
Older childYounger childP(this family)P(ALOB given this family)P(ALOG given this family)P(ALOB and this family)P(ALOG and this family)GirlGirl010GirlBoyBoyGirlBoyBoy100
So, if at least one is a boy when the fact is chosen randomly, the probability that both are boys is
The paradox occurs when it is not known how the statement "at least one is a boy" was generated. Either answer could be correct, based on what is assumed.
However, the "" answer is obtained only by assuming P(ALOB | BG) = P(ALOB | GB) = 1, which implies P(ALOG | BG) = P(ALOG | GB) = 0, that is, the other child's sex is never mentioned although it is present. As Marks and Smith say, "This extreme assumption is never included in the presentation of the two-child problem, however, and is surely not what people have in mind when they present it."
Modelling the generative process Another way to analyse the ambiguity (for question 2) is by making explicit the generative process (all draws are independent). The following process leads to answer : Draw equiprobably from Draw equiprobably from Discard cases where there is no B Observe The following process leads to answer : Draw equiprobably from Draw equiprobably from Draw index equiprobably from Observe
Bayesian analysis
Following classical probability arguments, we consider a large urn containing two children. We assume equal probability that either is a boy or a girl. The three discernible cases are thus:
both are girls (GG) – with probability P(GG) = , both are boys (BB) – with probability of P(BB) = , and one of each (G·B) – with probability of P(G·B) = .
These are the prior probabilities.
Now we add the additional assumption that "at least one is a boy" = B. Using Bayes' Theorem, we find
where P(A|B) means "probability of A given B". P(B|BB) = probability of at least one boy given both are boys = 1. P(BB) = probability of both boys = from the prior distribution. P(B) = probability of at least one being a boy, which includes cases BB and G·B = + = .
Note that, although the natural assumption seems to be a probability of , so the derived value of seems low, the actual "normal" value for P(BB) is , so the is actually a bit higher.
The paradox arises because the second assumption is somewhat artificial, and when describing the problem in an actual setting things get a bit sticky. Just how do we know that "at least" one is a boy? One description of the problem states that we look into a window, see only one child and it is a boy. This sounds like the same assumption. However, this one is equivalent to "sampling" the distribution (i.e. removing one child from the urn, ascertaining that it is a boy, then replacing). Let's call the statement "the sample is a boy" proposition "b". Now we have:
The difference here is the P(b), which is just the probability of drawing a boy from all possible cases (i.e. without the "at least"), which is clearly .
The Bayesian analysis generalizes easily to the case in which we relax the 50:50 population assumption. If we have no information about the populations then we assume a "flat prior", i.e. P(GG) = P(BB) = P(G·B) = . In this case the "at least" assumption produces the result P(BB|B) = , and the sampling assumption produces P(BB|b) = , a result also derivable from the Rule of Succession.
Variants of the question
Following the popularization of the paradox by Gardner it has been presented and discussed in various forms. The first variant presented by Bar-Hillel & Falk is worded as follows: 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? Bar-Hillel & Falk use this variant to highlight the importance of considering the underlying assumptions. The intuitive answer is and, when making the most natural assumptions, this is correct. However, someone may argue that "...before Mr. Smith identifies the boy as his son, we know only that he is either the father of two boys, BB, or of two girls, GG, or of one of each in either birth order, i.e., BG or GB. Assuming again independence and equiprobability, we begin with a probability of that Smith is the father of two boys. Discovering that he has at least one boy rules out the event GG. Since the remaining three events were equiprobable, we obtain a probability of for BB."