Probability that girl who answers the door is the eldest girl? You know that a family has 3 children. You walk up to and knock on the front door of their house. A girl answers the door. What is the probability that the she is the eldest girl among the children? Assume that all 3 children are home and equally likely to answer the door. (I appreciate that with questions like this, the wording can be so crucial. If the question is in any way ambiguous, I would be delighted to have an explanation of why that is the case!)

There are $8*3=24$ equally likely situations, the $8$ gender orderings and for each one, one of {youngest, middle, eldest} child who answers the door. $12$ of those have a girl answering the door (count the $12$ G's in the list given in the answer above).

Among those $12$ equally likely G's, exactly $7$ are the eldest girl (one from each of the $7$ non-BBB orderings). So the desired probability is $7/12$.

The condition "a random child turns out to be a girl" is NOT equivalent to a YES answer to the question "is there at least one girl?"