Probability of having a Girl A and B are married. They have two kids. One of them is a girl. What is the probability that the other kid is also a girl?
Someone says $\frac{1}{2}$, someone says $\frac{1}{3}$. Which is correct?
Now A and B have 4 children and all of them are boys. B is pregnant. So what is the probability that A and B are gifted with a baby girl?
Is it $\frac{1}{2}$ or there will be some conditional probability?

One of the kids is a girl does not convey enough information to answer the question. This is an example where we need to be extra careful using ordinary language to express what we mean mathematically.
Does it mean :


*

*One specific kid is a girl ( for example first born ).

*Any one of the two kids is a girl.

*Exactly one of the kids is a girl (thanks to drhab's comment)


For 1 we only have the freedom left to consider two cases girl & boy which should be an equi-probable independent event. That is: probability $1/2$.
For 2 it is like sampling one of the letters $\{BB, BG, GB, GG\}$, noticing that it is a $G$ and then wondering what the other would be. As you can see there is only one G which has another G besides it but three which have any 1 G, so here the probability it would be $1/3$.
For 3 it could be if a parent says "I have two children of which one is a girl" it probably is assumed that only one of the kids is a girl and then the probability of second kid to be a girl would be 0.
