  We estimate the probability of random $N$-qudit pure states violating
full-correlation Bell inequalities with two dichotomic observables per site.
These inequalities can show violations that grow exponentially with $N$, but we
prove this is not the typical case. For many-qubit states the probability to
violate any of these inequalities by an amount that grows linearly with $N$ is
vanishingly small. If each system's Hilbert space dimension is larger than two,
on the other hand, the probability of seeing \emph{any} violation is already
small. For the qubits case we discuss furthermore the consequences of this
result for the probability of seeing arbitrary violations (\emph i.e., of any
order of magnitude) when experimental imperfections are considered.
