# Probability: Die roll until first number

1. Jul 12, 2012

### stosw

1. The problem statement, all variables and given/known data

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?

2. Relevant equations

E[X] ?

3. 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.

2. Jul 12, 2012

### micromass

Staff Emeritus
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 $P\{X=n\}$?

3. Jul 12, 2012

### 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 $\sum_{k=1}^{\infty} P(X = 3k)$.