  We consider the zero distribution of random polynomials of the form $P_n(z) =
\sum_{k=0}^n a_k B_k(z)$, where $\{a_k\}_{k=0}^{\infty}$ are non-trivial i.i.d.
complex random variables with mean $0$ and finite variance. Polynomials
$\{B_k\}_{k=0}^{\infty}$ are selected from a standard basis such as Szeg\H{o},
Bergman, or Faber polynomials associated with a Jordan domain $G$ whose
boundary is $C^{2, \alpha}$ smooth. We show that the zero counting measures of
$P_n$ converge almost surely to the equilibrium measure on the boundary of $G$.
We also show that if $\{a_k\}_{k=0}^{\infty}$ are i.i.d. random variables, and
the domain $G$ has analytic boundary, then for a random series of the form
$f(z) =\sum_{k=0}^{\infty}a_k B_k(z),$ $\partial{G}$ is almost surely a natural
boundary for $f(z).$
