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Boy or girl paradox
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Boy or girl paradox
The Boy or Girl paradox surrounds a set of questions in probability theory, which are also known as The Two Child 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 October 1959 "Mathematical Games column" in Scientific American. He titled it The Two Children Problem and phrased the paradox as follows:
Gardner initially gave the answers 1/2 and 1/3, respectively, but later acknowledged that the second question was ambiguous. Its answer could be 1/2, 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 1/2 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 1/2. 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.
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 = 1/2:
Under the aforementioned assumptions, in this problem, a random family is selected. In this sample space, there are four equally probable events:
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 1/2.
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:
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Boy or girl paradox AI simulator
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Boy or girl paradox
The Boy or Girl paradox surrounds a set of questions in probability theory, which are also known as The Two Child 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 October 1959 "Mathematical Games column" in Scientific American. He titled it The Two Children Problem and phrased the paradox as follows:
Gardner initially gave the answers 1/2 and 1/3, respectively, but later acknowledged that the second question was ambiguous. Its answer could be 1/2, 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 1/2 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 1/2. 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.
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 = 1/2:
Under the aforementioned assumptions, in this problem, a random family is selected. In this sample space, there are four equally probable events:
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 1/2.
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:
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