  There has been considerable interest in recent decades in questions of random
generation of finite and profinite groups, and finite simple groups in
particular. In this paper we study similar notions for finite and profinite
associative algebras. Let $k=F_q$ be a finite field. Let $A$ be a finite
dimensional, associative, unital algebra over $k$. Let $P(A)$ be the
probability that two elements of $A$ chosen (uniformly and independently) at
random will generate $A$ as a unital $k$-algebra. It is known that, if $A$ is
simple, then $P(A) \to 1$ as $|A| \to \infty$. We extend this result to a large
class of finite associative algebras. For $A$ simple, we find the optimal lower
bound for $P(A)$ and we estimate the growth rate of $P(A)$ in terms of the
minimal index $m(A)$ of any proper subalgebra of $A$. We also study the random
generation of simple algebras $A$ by two elements that have a given
characteristic polynomial (resp. a given rank). In addition, we bound above and
below the minimal number of generators of general finite algebras. Finally, we
let $A$ be a profinite algebra over $k$. We show that $A$ is positively
finitely generated if and only if $A$ has polynomial maximal subalgebra growth.
Related quantitative results are also established.
