<p>We consider strategy-proof social choice correspondences (SCCs) –mappings from preference profiles to sets of alternatives– when individuals are endowed with single-peaked preferences over alternatives. We interpret the selected sets of alternatives as the basis for lotteries that determine the final social choice, and consider that agents’ preferences over sets are consistent with Expected Utility Theory and Bayesian updating from an initial probability assessment over the full set of alternatives. We exploit the relation between SCCs and probabilistic decision schemes –mappings from preference profiles to lotteries over alternatives–, to characterize the family of SCCs that satisfy strategy-proofness and unanimity for arbitrary initial probability assessments. We extend the analysis to multi-dimensional convex spaces of alternatives under the uniform initial probability assessment.</p>