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,[1] Mr. Smith's Children[2] 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:

  • 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 ⁠1/2⁠ and ⁠1/3⁠, respectively, but later acknowledged that the second question was ambiguous.[1] 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,[3] and Raymond S. Nickerson.[4]

Other variants of this question, with varying degrees of ambiguity, have been popularized by Ask Marilyn in Parade Magazine,[5] John Tierney of The New York Times,[6] and Leonard Mlodinow in The Drunkard's Walk.[7] 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%.[2]

The paradox has stimulated a great deal of controversy.[4] The paradox stems from whether the problem setup is similar for the two questions.[2][7] The intuitive answer is ⁠1/2⁠.[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),[2] and that the probability of these outcomes is absolute, not conditional.[8]

The two interpretations of the second part are shown in 2a and 2b, the probability in each case being the fraction of the shaded cell to the outlined ones.
  1. ^ a b Martin Gardner (1961). The Second Scientific American Book of Mathematical Puzzles and Diversions. Simon & Schuster. ISBN 978-0-226-28253-4.
  2. ^ a b c d e Cite error: The named reference fox was invoked but never defined (see the help page).
  3. ^ Cite error: The named reference Bar-Hillel and Falk was invoked but never defined (see the help page).
  4. ^ a b Raymond S. Nickerson (May 2004). Cognition and Chance: The Psychology of Probabilistic Reasoning. Psychology Press. ISBN 0-8058-4899-1.
  5. ^ "Ask Marilyn". Parade Magazine. October 13, 1991 [January 5, 1992; May 26, 1996; December 1, 1996; March 30, 1997; July 27, 1997; October 19, 1997].
  6. ^ Tierney, John (2008-04-10). "The psychology of getting suckered". The New York Times. Retrieved 24 February 2009.
  7. ^ a b Leonard Mlodinow (2008). The Drunkard's Walk: How Randomness Rules our Lives. Pantheon. ISBN 978-0-375-42404-5.
  8. ^ P.J. Laird; et al. (1999). "Naive Probability: A Mental Model Theory of Extensional Reasoning". Psychological Review. 106 (1): 62–88. doi:10.1037/0033-295x.106.1.62. PMID 10197363.