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?

Conditional probability:
Let $A$ and $B$ be two events.
$$P(A|B) = \frac{P(A\cap B)}{P(B)}$$
which means, the probability of $A$ occuring given that $B$ occured, is the probability of both $A$ and $B$ occuring, divided by the probability that $B$ occurs.
In this case, $A$ is the event that the other kid is a girl, and $B$ is the event that one of the kids is a girl.
$A\cap B$ would be both kids are girls, which has a probability of $\frac14$.
$B$ would be the event that one of the kids is a girl, which has a probability of $\frac34$.
Therefore, the required probability is $\frac{1/4}{3/4}=\frac13$.

And for the second question, we can do it using conditional probability or without.
Using conditional probability:
$A$ be the event that the fifth child is a girl.
$B$ be the event that the first four children are boys.
$P(A\cap B)$ would be $\left(\frac12\right)^5=\frac1{32}$.
$P(B)$ would be $\left(\frac12\right)^4=\frac1{16}$.
The required probability would be $\frac{1/32}{1/16}=\frac12$.
Without using conditional probability:
Notice that event $A$ is independent on event $B$.
Therefore, $P(A)=\frac12$.

Notes
In the first question, the probability in question is the probability that "the other kid is also a girl", making the two events dependent.
However, in the second question, the word other is gone, leaving us with independent events.
