This is a question I originally posted on Math Stackexchange. I've just seen a very good discussion of Monty Hall problem, and someone mentioned the "siblings" paradox. I've had some success explaining the Monty Hall, but ran into some tough resistance in the siblings paradox. A summary of the arguments I faced is below.
So here is the problem:
There is a well-known paradox (mentioned, among other sources, by Martin Gardner).
A) You know that I have two children, and the oldest is a daughter. What is the probability that both are daughters? (let's ignore the possibility of identical twins, which may muddle the question; and, for simplicity, assume that probability that a child is a daughter is exactly 1/2). This is very easy to answer.
B) Now, suppose you know that at least one child is a daughter (but you don't know if it's the oldest one or not). What is the probability that both are daughters?
The common way to answer is: you got extra information that the number of daughters, let's call it D, is at least one. Thus, $P(D=2 | D \ge 1)= P(D=2,D \ge 1)/P(D \ge 1)=(1/4)/(3/4)=1/3$.
Another way to answer (B) is to consider all possible outcomes.
The paradox is that the answers in (A) and (B) are different!
However, a student told me that this is a wrong way to look at it. How do you know that at least one child is a daughter? Perhaps you overheard me saying "my daughter ... (just had a birthday or something)". But in this case you were more likely to overhear in the first place, if I had had two daughters, and not one. This messes up the probabilities calculated above.
To put it another way: let's generate the list of random 2-child families. Let's randomly "overhear" the gender of one random child in each family on the list. [This is what I foolishly suggested the student do!] Ignore sons and concentrate on daughters. Calculate which proportion of the families we picked had two daughters. Then your answer is 1/2 again.
Can you describe a realistic situation in which you can come up with the information "at least one girl", and obtain the "right" answer of 1/3?