  We show that the density of quadratic forms in $n$ variables over $\mathbb
Z_p$ that are isotropic is a rational function of $p$, where the rational
function is independent of $p$, and we determine this rational function
explicitly. When real quadratic forms in $n$ variables are distributed
according to the Gaussian Orthogonal Ensemble (GOE) of random matrix theory, we
determine explicitly the probability that a random such real quadratic form is
isotropic (i.e., indefinite).
  As a consequence, for each $n$, we determine an exact expression for the
probability that a random integral quadratic form in $n$ variables is isotropic
(i.e., has a nontrivial zero over $\mathbb Z$), when these integral quadratic
forms are chosen according to the GOE distribution. In particular, we find an
exact expression for the probability that a random integral quaternary
quadratic form has an integral zero; numerically, this probability is
approximately $98.3\%$.
