I am having trouble understanding how the probability of at least one girl is 3/4 in this question below.
Given a family of two children (assume boys and girls equally likely, that is, probability 1/2 for each), That at least one
is a girl?
My reason for the confusion is that, if at least one is a girl then the probability of having a boy-boy does not exist, therefore the sample space becomes
[GG,BG,GB]
so surely the answer should be 2/3
Best Answer
Your sample space is not $[GG, BG, GB].$
This gives us (so far) the sample space $[GG, BG, GB, BB].$
You left these words out of the question, but I think they were implied. They mean that we should now get ready to compute the probability of some event.
OK, so the event is "at least one girl." The following elements of your sample space have at least one girl: $GG, BG, GB.$ The element $BB$ does not.
So the event we are asked to give the probability of is $[GG, BG, GB].$
The sample space is still $[GG, BG, GB, BB].$
We are evidently meant to assume that the births are independent as well as equally likely to be boys or girls, so the elements of the sample space are equally likely. Thus an event with $3$ of the $4$ elements has probability $3/4.$