The Difference Between the Two Definitions of the Geometric Distribution
Explanation
While studying the geometric distribution, the most confusing and bewildering thing is that the explanation differs from textbook to textbook, blog to blog, and wiki to wiki. In some places the mean is $\displaystyle {{1} \over {p}} $, while in others it is written as $\displaystyle {{1-p} \over {p}}$.
This difference arises because there are two ways to define the geometric distribution. The probability mass function of the geometric distribution $\text{Geo}(p)$ is defined as $$ p_{1}(x) = p(1-p)^{x-1} , x= 1,2,3,\cdots $$ or $$ p_{2}(x) = p(1-p)^{x} , x= 0,1,2,\cdots $$ The expected value is determined by the probability mass function, so using $p_{1}$ gives $\displaystyle {{1} \over {p}}$, while using $p_{2}$ gives $\displaystyle {{1-p} \over {p}}$.
As you can see by looking closely at the probability mass functions, there is no essential difference between the two definitions; ultimately it comes down to whether we start from $1$ or from $0$. Thinking about the intuitive definition of the geometric distribution, if we call the occurrence of some event a ‘success’, there are only two options: whether we are interested in the number of trials until success, or the number of failures until success. If we succeed on the first try, the number of trials would be $1$ and the number of failures would be $0$.
Meanwhile, since the geometric distribution has the memorylessness property, it can be used in survival analysis, and if we call the occurrence of an event ‘breakdown’, what we are interested in would be how many times something endures without breaking. In such a case, counting the number of ‘failures’ mentioned above becomes meaningful.
Ultimately, which probability mass function you choose can be regarded as being determined by the object of interest, convenience, convention, and so on. Do not overthink it; just think of it as people simply using whatever they want to use, and move on.