Answer
$\dfrac{1}{6}$
Work Step by Step
Here, $A$ denotes the red die is $4$ and $B$ denotes the green die is $4$.
Our aim is to calculate the conditional probability $P(A|B)$.
This can be found as: $P(A|B)=\dfrac{P(A \cap B)}{P(B)} ~~~~(1)$
We can see that $A \cap B$ shows of all outcomes that red die is $4$ and the green die is $4$. This means that $A \cap B=\{(4,4)\}$
So, $P(A \cap B)=\dfrac{n(A \cap B)}{n(S)}=\dfrac{1}{36}$
and $P(B)=\dfrac{n(B)}{n(S)}=\dfrac{6}{36}$
Thus, the equation (1) becomes:
$P(A~|~B)=\dfrac{P(A \cap B)}{P(B)} =\dfrac{1/36}{6/36}=\dfrac{1}{6}$