Variance problem

Question

So I know how to calculate variance, but I ran into a peculiar problem. If I roll a dice until I get a 5 or 6, what would the variance of this distribution be? I've also been told the answer is 6 but I have no idea how to start this problem.

Answer 1

Let random variable $X$ be the number of trials (tosses) until we get a $5$ or a $6$. Then $X$ has geometric distribution with parameter $p=\frac{1}{3}$. (Here $p$ is the probability of "success" on any one trial.)

The variance of a geometric distribution with parameter $p$ is $\frac{1-p}{p^2}$. When $p=\frac{1}{3}$, the variance of $X$ is indeed $6$.