I am currently taking a probability course and I am stuck on a supposedly easy discrete probability question here:
Problem: Consider the experiment of rolling a fair die independently until the same number/face occurs 2 successive times and let $X$ be the trial on which the repeat occurs, e.g. if the rolls are $2,3,4,5,1,2,4,5,5$, then $X=9$.
a. find the probability function $f(x) = P(X=x)$
b. compute EX
Attempt at a solution: I know $X$ is discrete probability distribution, and that we are dealing with independent events. However, $X$ can be anything, up to infinity, or it may never happen where there are two successive values. Here's what I got:
obviously, the answer is a geometric distribution as the answer is:
$P(X=x) = f(x) = (5/6)^(x-1) * (1/6)$ for $x = 0,1,2,3,...$ and $0$ otherwise but I'm stuck here. please help