Here's an issue I was told some years ago by a maths professor, and I can't figure out what is actually going on, so I wanted to ask you if you can help me. Imagine there is family from which you know that it has two children, but you can't recall their gender. Both genders, male and female, shall be equally probable.
1) You are told that one of the children is female. What is the probability that the other is male? Answer: There are four possibilities: First child male, second child male (MM), first child male, second child female (MF), FM, FF. There first one is ruled out by the assumption that one is female, the others are equally probable. As two of them contain a male and one a female, the probability that the other is male is 2/3 (and not 1/2, as one might think).
2) Now it is getting pretty weird: You are told that one is female and born on a monday. What is the probability now that the other one is male? Answer: Now, there are 14*14 possibilities: First child female born on monday, second child female born on monday (F1F1); F1F2; F1F3;...; F1F7; F1M1;...; F1M7; F2F1;...; F2M7; F3F1;...; F7M7; M1F1;...; M7M7. Every possibility without a female child on a monday is ruled out, so there are 27 remaining; F1F1, ... F1M7,F2F1,...,M7F1. 14 of these contain a male child, 13 a female, and all are equally probable, so the final probability is 14/27 ???
3) (That one is by me). Imagine you are given that one of the children is female and the exact time of birth, e.g. 10am, 57 minutes, 12 seconds,...; up to some interval Dt. If you take Dt-> 0, the probability should get arbitrarily close to 1/2
I know thats all pretty messed up. Do you have any idea on how to fix it?