The following two problems appeared together in Martin Gardner’s Scientific American column in 1959.
Mr. Smith has two children. At least one of them is a boy. What is the probability that both children are boys?
Mr. Jones has two children. The older child is a girl. What is the probability that both children are girls?
Many people, including me and Martin Gardner, wrote a lot about Mr. Smith. In his original column Martin Gardner argued that the answer to the first problem is 1/3. Later he wrote a column titled “Probability and Ambiguity,” where he corrected himself about Mr. Smith.
… the answer depends on the procedure by which the information “at least one is a boy” is obtained.
This time I would like to ignore Mr. Smith, as I wrote a whole paper about him that is now under consideration for publication at the College Mathematics Journal. I would rather get back to Mr. Jones.
Mr. Jones failed to stir a controversy from the start and was forgotten. Olivier Leguay asked me about Mr. Jones in a private email, reminding me that the answer to the problem about his children also depends on the procedure.
One of the reasons Mr. Jones was forgotten is that for many natural procedures the answer is 1/2. For example, the following procedures will produce an answer of 1/2:
- We ask Mr. Jones whether his older child is a daughter and he says “yes.”
- Mr. Jones flips a coin deciding which child to talk about. After that he has to tell us the gender and whether the child is the oldest.
- Mr. Jones is asked to say nothing if he doesn’t have a daughter, to select the daughter if has just one, or to pick one at random if he has two daughters. After that he has to tell us whether the daughter he has selected is the oldest.
There are many other procedures that lead to the answer 1/2. However, there are many procedures that lead to other answers.
Suppose I know Mr. Jones, and also know that he has two children. I meet Mr. Jones at a mall, and he tells me that he is buying a gift for his older daughter. Most probably I would assume that the other child is a daughter, too. In my experience, people who have a son and a daughter would say that they are buying a gift for “my daughter.” Only people with two daughters would bother to specify that they are buying a gift for “my older daughter.”
In some sense I didn’t forget about Mr. Jones. I wrote about him implicitly in my essay Two Coins Puzzle. His name was Carl and he had two coins instead of two children.Share:
Martin Gardner wrote an article about two hypothetical gentlemen. If at least one of them stirred up controversy, what is the probability that both of them did?19 February 2011, 4:54 pm
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Regardless of what is said, you can only ever conclude the probability that his other child is male or female to be 1/2 because childbirth/gender is independent. I can’t tell you how many times I’ve meant to say “Desktop” or “Primary Computer” and have instead said “Primary Desktop”, which indicates I have more than one desktop, which I don’t… People most places look at me liek I’m made of money, which I’m not.. so it doesn’t matter one iota how the information was obtained, so we should assume that all parents with 2 children are collected, and ones meeting certain criteria are chosen. the rest are summarily executed without their child genders being recorded for statistical analysis. we then must surmise the probability that the one unknown child’s gender is either male or female.21 February 2011, 2:17 pm