  We study coalitional games with exogenous uncertainty in the coalition value,
in which each agent is allowed to have private samples of the uncertainty. As a
consequence, the agents may have a different perception of stability of the
grand coalition. In this context, we propose a novel methodology to study the
out-of-sample coalitional rationality of allocations in the set of stable
allocations (i.e., the core). Our analysis builds on the framework of probably
approximately correct learning. Initially, we state a priori and a posteriori
guarantees for the entire core. Furthermore, we provide a distributed algorithm
to compute a compression set that determines the generalization properties of
the a posteriori statements. We then refine our probabilistic robustness bounds
by specialising the analysis to a single payoff allocation, taking, also in
this case, both a priori and a posteriori approaches. Finally, we consider a
relaxed $\zeta$-core to include nearby allocations and also address the case of
empty core. For this case, probabilistic statements are given on the eventual
stability of allocations in the $\zeta$-core.
