Binomial Distribution: Finding expected number of trials

Question

Suppose, the player is rolling a die and wins only when gets a multiply of 3 (3 or 6) 4 times. How can I calculate the expected number of rolls aka the number of rolls after which the die has shown the multiply of 3 for 4 times?

Answer 1

$E=p\sum\limits_{n=0}^\infty (n+4) \binom{n+3}{3}q^np^3$, where $p=\frac{1}{3}$ is the probability of getting a multiple of $3$ and $q=1-p$.

Explanation. Leading $p$ is fourth successful roll probability. $\ n+4$ is total number of rolls. $\ q^n$ is probability of getting $n$ bad rolls, while $p^3$ is probability of getting $3$ good rolls. $\ \binom{n+3}{3}$ is the number of possibilities of these $(n+3)$ rolls.