For $0<p<\infty$, the Bergman space $A^p$ consists of functions $f$ analytic in the unit disk $\Bbb D$ such that

$$\int_{\Bbb D}|f(z)|^p\,dA(z)<\infty,$$

where $dA(z)$ is the Lebesgue area measure on $\Bbb D.$ We say that a sequence of points $\{z_n\}$ in $\Bbb D$ is an $A^p$ zero set if some function $f\in A^p$ vanishes precisely on this set. Many interesting results about zero sets for Bergman spaces have been obtained by Horowitz [Ho1, Ho2], Korenblum [K] and Seip [S]. The reader is also referred to [HKZ]. The probabilistic approach to the study of $A^p$ zero sets was initiated by Leblanc [LeB] who proved a sufficient condition for a sequence $\{z_n\}$ to be an $A^2$ zero set almost certainly when the arguments of $z_n$ are chosen at random. His results have been improved and extended to $0<p\le 2$ by Bomash [Bo].