What is the probability that a random integral quadratic form in $n$ variables is isotropic?

We show that the density of quadratic forms in $n$ variables over ${\mathbb Z}_p$thatareisotropicisarationalfunctionin$p$, 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 $\approx 97.0\mathrm{\%}$, in the case where the coefficients of the quadratic form are independently and uniformly distributed in the range $[-X,X]$ with $X\to \mathrm{\infty }$. When random integral quaternary quadratic forms are chosen with respect to the Gaussian Orthogonal Ensemble (GOE), the probability of isotropy increases to $\approx 98.3\mathrm{\%}$.