  Theoretical considerations of Bell-inequality experiments usually assume
identically prepared and independent pairs of particles. Here we consider pairs
that exhibit both intra- and inter-pair entanglement. The pairs are taken from
a large many-body system where all the pairs are generally entangled with each
other. Using an explicit example based on single mode entanglement and an
ancillary Bose-Einstein condensate, we show that the Bell-inequality violation
in such systems can display statistical properties that are remarkably
different from those obtained using identically prepared, independent pairs. In
particular, one can have probabilistic violation of Bell's inequalities in
which a finite fraction of all the runs result in violation, even though there
could be no violation when averaging over all the runs. Whether or not a
particular run of results will end up being local realistically explainable is
"decided" by a sequence of quantum (random) outcomes.
