TITLE: In Geometric distribution what does it mean to include first success
QUESTION [2 upvotes]: I was watching the harvard stat110 course on Geometric distribution and the PMF is $P(X = k) = pq^k$ where $p$ is the probability of success and $q = 1-p$. later he mentions another r.v $Y$ which he defined as the 1st success distribution and he mentions that this distribution, unlike the Geometric Distribution, includes the first success.
I got a bit confused as to the meaning of including the first success, in the PMF of $Geo(p)$ there is a $p$, isn't this including the first success already?

REPLY [3 votes]: There are two common forms of the geometric distribution.
Imagine we have a sequence of Bernoulli trials (specifically, imagine we're tossing a biased coin).
I'm interested in $Y$, the number of trials to get the first head (how many tosses did I make to get a head), so let's call "got a head" a success, and it's the number of trials to get the first success. If I toss T, T, H, I count 3 trials. 
My sister is interested in the number of times I failed to get a head before I succeeded, $X$ (i.e. she only cares how many tails I got). If I toss T, T, H she counts "two failures".
Both these variables are called "geometric" and the only difference is that my variable counts the trial on which I got a success in the total while my sisters variable does not (i.e. $Y=X+1$)
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