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

Small violations of full correlation Bell inequalities for multipartite pure random states