Let $A$ be a random $n \times n$ matrix such that $A_{ij}\in\{0,1\}$. Assume that each element $A_{ij}$ equals 1 with some probability $p>0$ and that all the draws are independent across elements. What is the probability that the symmetric part of $A$, the matrix $\frac{1}{2}(A+A^T)$, is positive semi-definite?
Any results pertaining to the symmetric part being positive definite would also be welcome.