  In this paper, we investigate the following question: How often is a random
matrix normal? We consider a random $n\times n$ matrix, $M_n$, whose entries
are i.i.d. Rademacher random variables (taking values $\{ \pm1 \}$ with
probability $1/2$) and prove
  $$2^{-\left(0.5+o(1)\right)n^2} \le P\left(M_n \text{ is normal}\right) \le
2^{-(0.302+o(1))n^{2}}. $$ We conjecture that the lower bound is sharp.
