A couple has 2 children. What is the probability that both are girls if the eldest is a girl?

Question

This is another question like this one. And by the same reason, the book only has the final answer, I'd like to check if my reasoning is right.

A couple has 2 children. What is the probability that both are girls if the eldest is a girl?

Answer 1

The sample space is $S = \{(g,g),(g,b),(b,g),(b,b)\}$, where $b$ is for boy, $g$ is for girl the first element of the tuple is the eldest.

Let $B$ the event the eldest is a girl, so $B=\{(g,b),(g,g)\}$.

$A$ is the event where the two children are girls. $A = \{(g,g)\}$.

Then:

$$ P(A|B)=\frac{P(A\cap B)}{P(B)}=\frac{\dfrac{|A\cap B|}{|S|}}{\dfrac{|B|}{|S|}}=\frac{1}{2}$$.

The end.

Answer 2

An alternative viewpoint:

For the eldest child to be a girl, they must have had a girl first. Therefore the probability of there being two girls is the probability of having a second girl which is $\frac{1}{2}$.