Probability: Die roll until first number

[stosw]

Homework Statement

Roll a fair die until the first time you roll a '5' then stop. Let X be the number of times you rolled the die. What is the probability that X is divisible by 3?

Homework Equations

E[X] ?

The Attempt at a Solution

I honestly have no idea how to even approach this sort of thing. My first thought was either a bionomial or geometric series, but that could possibly be endless.

any sort of hints or suggestions would be great.

 

[micromass]

Let's first start by figuring out the probabilities involved.

Can you find out $P\{X=1\}$? (that is: the probability your first throw is a 5). What about $P\{X=2\}$? (the probability that your first throw is not a five, but your second throw is). Can you find a general formula for $P\{X=n\}$?

 

[QED Andrew]

Find $P(X = x)$. $X$ is divisible by $3$ if and only if $X = 3k$ for some integer $k$. Therefore, the probability that $X$ is divisible by $3$ is given by $\sum_{k=1}^{\infty} P(X = 3k)$.