The Bayesian analysis generalizes easily to the case in which we relax the If we have no information about the populations then we assume a "flat prior", i. Martingale analysis[ edit ] Suppose you had wagered that Mr Smith had two boys, and received fair odds. We think of your wager as investment that will increase in value as good news arrives.
What evidence would make you happier about your investment? Learning that at least one child out of two is a boy, or learning that at least one child out of one is a boy? The latter is a priori less likely, and therefore better news. That is why the two answers cannot be the same.
Now for the numbers. If we bet on one child and win, the value of your investment has doubled. On the other hand if we learn that at least one of two children is a boy, our investment increases as if we had wagered on this question. So the answer is 1 in 3. Variants of the question[ edit ] Following the popularization of the paradox by Gardner it has been presented and discussed in various forms. 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? 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. Discovering that he has at least one boy rules out the event GG.
Smith selected the child companion at random. They imagine a culture in which boys are invariably chosen over girls as walking companions. In , Marilyn vos Savant responded to a reader who asked her to answer a variant of the Boy or Girl paradox that included beagles. The and questions, respectively were phrased: A shopkeeper says she has two new baby beagles to show you, but she doesn't know whether they're male, female, or a pair. You tell her that you want only a male, and she telephones the fellow who's giving them a bath.
What is the probability that the other one is a male? Say that a woman and a man who are unrelated each has two children. We know that at least one of the woman's children is a boy and that the man's oldest child is a boy. Can you explain why the chances that the woman has two boys do not equal the chances that the man has two boys?
In response to reader response that questioned her analysis vos Savant conducted a survey of readers with exactly two children, at least one of which is a boy. Of 17, responses, The authors do not discuss the possible ambiguity in the question and conclude that her answer is correct from a mathematical perspective, given the assumptions that the likelihood of a child being a boy or girl is equal, and that the sex of the second child is independent of the first.
They demonstrate that in reality male children are actually more likely than female children, and that the sex of the second child is not independent of the sex of the first. The authors conclude that, although the assumptions of the question run counter to observations, the paradox still has pedagogical value, since it "illustrates one of the more intriguing applications of conditional probability.
Information about the child[ edit ] Suppose we were told not only that Mr. Smith has two children, and one of them is a boy, but also that the boy was born on a Tuesday: Again, the answer depends on how this information was presented - what kind of selection process produced this knowledge.
Following the tradition of the problem, suppose that in the population of two-child families, the sex of the two children is independent of one another, equally likely boy or girl, and that the birth date of each child is independent of the other child. From Bayes' Theorem that the probability of two boys, given that one boy was born on a Tuesday is given by: