Wasserstein distances provide a metric on a space of probability measures. We consider the space $Ω$ of all probability measures on the finite set $χ= \{1, \dots ,n\}$ where $n$ is a positive integer. 1-Wasserstein distance, $W_1(μ,ν)$ is a function from $Ω\times Ω$ to $[0,\infty)$. This paper derives closed form expressions for the First and Second moment of $W_1$ on $Ω\times Ω$ assuming a uniform distribution on $Ω\times Ω$.