Dice and probability

Question

Two fair six-sided dice are rolled. If the sum of the numbers obtained is $4$, find the probability that the numbers obtained on both dice are even.

I'm not sure if it's $\frac{1}{36}$ or $\frac{1}{3}$.

Answer 1

$4$ can obtained by $1+3,2+2,3+1$

So, the probability $\displaystyle=\frac{\text{No. of favorable cases}}{\text{No. of possible cases}}=\frac13$

Answer 2

The important point is that you are first given that the sum is $4$.

This can only happen rolling $(1,3), (2,2)$ or $(3,1)$.

So given that the sum is $4$, we now have a sample size of $3$, out of which only $1$ entry, i.e., $(2,2)$, has both dice showing an even number.

Therefore, the answer is $1/3$.

---

If, however, you were just asked: What is the chance of rolling $(2,2)$?

Then your sample size would be all the possible rolls with two dice, and you would see that only $1$ of the $36$ possibilities is $(2,2)$. In this case, different from yours, the answer would be $1/36$.