1) A couple has two children and at least one of them is female. What is the probability of the second child also being a female ?
2) A couple has one child, a second child is coming, what is the probability of this second child being female ?
3) A couple has one child, what is the probability of this child being a female ?
What is the difference between the interpretation of the questions above ? Why is the birth of a person influenced the birth of another ? Shouldn't they be statistically independent ? What is the correct interpretation ? For me, intuitively, the answer for the probability of being a female or male is always 50%, independent of the above interpretation.
Extra: 4) What is the probability of a couple having two female children ? Now, for me, it is clear that the probability is 25%. What is the correct interpretation of the questions above ?
Thanks.
The first one is a conditional probability. What is the probability that both children are girls, given that at least one of them is a girl. You seem to be thinking that the question says something like, "The older one is a girl," but that's not correct.
The second is what you say. The probability is $\frac12$.
The third is the same.
For the fourth, I don't see how it can be answered as written. We don't even know the probability that a couple has a least two children. The answer you give $\frac14$ is correct, given that the couple has exactly two children, but the question doesn't say that.
EDIT
The OP points out correctly, that the second one is a conditional probability also. However in that case, the second event is independent of the first. The probability that the second child is a girl given that the first child is a girl is the same as the probability that the second child is a girl, which we assume to be $\frac12$.
In the first case, the condition really matters. We have a priori, four equally likely possibilities: BB, BG, GB, GG, and the condition eliminates the first of these.