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Google words…Two Children Puzzle, Other Child Puzzle, Ask Marilyn Child Puzzle, etc. Last spring when this subject first came up, the first bunch of Google results were from colleges talking about this puzzle. They don’t appear now high on the list so they very well could have been course related. Wait a few weeks and they may appear again as we are at the start of the school year. But even today I found these in less than 15 minutes.
Ask Dr. Math(answered by Dr. TWE?)
Question: From the set of all families with two children, a child is
selected at random and is found to be a boy. What is the
probability that the other child of the family is a girl?
Answer: ½
Question: From the set of all families with two children, a family is
selected at random and is found to have a boy. What is the
probability that the other child of the family is a girl?
Answer: 2/3
*Why the difference?
As in other probability problems, how information is obtained is as
important as the information itself. Without knowledge of the data
gathering process, ambiguity can result. How do we know that one child
is a boy?
*
Ask Dr. Math)
From Tanya Khovanova PhD in Mathematics
Her question: A man says, “I have two children; at least one of them is a boy.” What is the probability that the other one is a boy?
Her answer: *The correct answer is that the problem is undefined. For example, there is no indication in the problem that all fathers with at least one boy will tell you, “I have at least one boy.”
*
TanyaKhovanova PhD
From Robert McMullen, author of Stranger and Stranger
His question: Professor Saunders has two children. At least one of them is a boy.
What is the probability that both his children are boys?
His answer: The answer is that the problem is not answerable without more information.
Robert McMullen
Not truly scholarly but….
Wikipedia….Brain Teaser
Question: *If someone has two children, and one of them is a son, what is the probability that the other is also a son?
….but that would be (more) ambiguous, since it could mean that we chose a person at random, and learnt that at least one of their two children was a son (in which case we get 1/3), or it could mean that we chose a person at random, and met one of their children, which turned out to be a son. This would then be a particular child, so the probability of the other being a son is 1/2.*
Wiki-Brain Teaser
As I have posted before, Scientific American and Martin Gardner have had almost 50 years to retract their statements, but as of yet they haven’t. And lastly, check Ask Marilyn’s website. She once said that between the two child puzzle and the Monty Hall puzzle, she believed every mathematician in America had written her. Do you really think they all wanted to pat her on the back?
For JR Brown,
“Many readers correctly pointed out that the answer depends on the procedure by which the information “at least one is a boy” is obtained. If from all families with two children, at least one of whom is a boy, a family is chosen at random, then the answer is 1/3. But there is another procedure that leads to exactly the same statement of the problem. From families with two children, one family is selected at random. If both children are boys, the informant says “at least one is a boy.” If both are girls, he says “at least one is a girl.” And if both sexes are represented, he picks a child at random and says “at least one is a …” naming the child picked. When this procedure is followed, the probability that both children are of the same sex is clearly 1/2. (This is easy to see because the informant makes a statement in each of the four cases – BB, BG, GB, GG – and in half of these case both children are of the same sex.) That the best of mathematicians can overlook such ambiguities is indicated by the fact that this problem, in unanswerable form, appeared in one of the best of recent college textbooks on modern mathematics.”
**Scientific American, October, 1959
**
I beleive that the magazine made it clear that if we don’t know how the information was obtain, the puzzle is ambigious. It is my opinion that we have to use a natural approach to this puzzle… we meet a couple (or are told of a couple) and that through conversation find out they have a two child and one of which is a girl. In the nature approach the family is random and the child is random. This version is easily repeatable as the only condition that exist is that the family has to have two children. In the 2/3 version we must always have a family with girl, which would require us to alway ask “do you at least one girl”, to screen the families. You cannot allow the family with both to say “I have a boy.”
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Okay, thanks for the links.
I think they’re wrong, but you’re right that these are experts and what they say has weight.
You are right that if we don’t know how the information was obtained, then the puzzle is ambiguous. But I maintain we do know how it was obtained. We don’t need to be told, because the phrasing “at least one child is a girl” implies a method by which the information was obtained. Absent specification, there is a “default” method which any native speaker of English would understand. “At least one child is a girl” means someone looked at the pair, determined whether at least one child is a girl, and reported to me that, yes, at least one is a girl.
To wonder, as Gardner et al do, how the information was obtained, is unnecessary, in the context of a puzzle like this.
-FrL-
and I therefore can’t look at Martin Gardner’s original problem, but the follow-up article you quote contains the sentence “Many readers correctly pointed out that the answer depends on the procedure by which the information “at least one is a boy” is obtained.” Therefore I assume the original problem contained the phrase “at least one is a boy”.
, but I stand by the assertion that the most likely interpretation of “You have been told this family has a daughter” is that the family was asked “Do you have any daughters?”. In which case the answer is 2/3.