  Let $G$ be an algebraic group. For $d\geq 1$, we define the commuting
probabilities $cp_d(G) = \frac{dim(\mathfrak C_d(G))}{dim(G^d)}$, where
$\mathfrak C_d(G)$ is the variety of commuting $d$-tuples in $G$. We prove that
for a reductive group $G$ when $d$ is large, $cp_d(G)\sim \frac{\alpha}{n}$
where $n=\dim(G)$, and $\alpha$ is the maximal dimension of an Abelian subgroup
of $G$. For a finite reductive group $G$ defined over the field $\mathbb F_q$,
we show that $cp_{d+1}(G(\mathbb F_q))\sim q^{(\alpha-n)d}$, and give several
examples.
