On 6/12/06, James Propp <propp@math.wisc.edu> wrote:
"A family has two children, at least one of whom is a boy; what is the probability that both are boys?"
When I teach this material, I tell the students that one standard version of the question -- "A family has two children, one of whom is a boy; what is the probability that the other one is a boy?" -- is too incoherent to have a clear answer. The first half of the question seems to be saying "at least one of the children is a boy", but if so, then the second half commits the fallacy of misplaced concreteness by talking about a boy in particular.
Yes, I also have issues with this phrasing. My usual way of presenting these problems is as a dialogue, something like A: I have two kids. B: Oh, do you have at least one boy? A: Yes. Of course nobody would ever ask the question that way, or answer it that way, in real life. The dialogues would go more like B: Do you have a boy? A: Yes, one. or B: Do you have a boy? A: Yes, two. But those dialogues don't make good probability problems. Nowadays, teaching more statistics than probability, I try to find good two-way tables that can lead to similar conditional probability complications while being more realistic, like "A man survived the Titanic. What is the probability that his wife survived?" where you have to figure out what class passenger he is likely to be, based on data about % survival for male and female adults by class, and worrying about assumptions that are more relevant to real-life uses of probability and statistics. I also like the examples about number of siblings -- I do that with my classes all the time, ask each kid how many siblings they have, and then ask statistically how far off the true mean number of siblings this sample is likely to be, and then point out that we have a simple random sample of KIDS which is far from a random sample of FAMILIES! That shows my students a lot about how tricky it can be to assemble a truly random sample of what they want to sample. --Joshua Zucker