  We show that the density of quadratic forms in $n$ variables over ${\mathbb
Z}_p$ that are isotropic is a rational function in $p$, where the rational
function is independent of $p$, and we determine this rational function
explicitly. As a consequence, for each $n$, we determine the probability that a
random integral quadratic form in $n$ variables is isotropic. In particular, we
show that the probability that a random integral quaternary quadratic form is
isotropic is $\approx 97.0\%$, in the case where the coefficients of the
quadratic form are independently and uniformly distributed in the range
$[-X,X]$ with $X\to\infty$. When random integral quaternary quadratic forms are
chosen with respect to the Gaussian Orthogonal Ensemble (GOE), the probability
of isotropy increases to $\approx 98.3\%$.

Number Theory