  We investigate the probability that a random quadratic form in
${\mathbb{Z}}[x_1,...,x_n]$ has a totally isotropic subspace of a given
dimension. We show that this global probability is a product of local
probabilities. Our main result computes these local probabilities for quadratic
forms over the $p$-adics. The formulae we obtain are rational functions in $p$
invariant upon substituting $p \mapsto 1/p$.
