Abstract: An accessible argument is given for why some correlations between quantum systems boggle our classical intuition.  The argument relies on simple properties of joint probabilities, and recovers the standard experimentally-testable Bell inequality in a form that applies equally well to correlations between six-sided dice and between photon polarizations. The observed violation of this inequality implies that the quantities measured on one system cannot have a joint probability distribution that is invariant with respect to the choice of measurement made on a distant system.   The possible but extraordinary physical mechanisms underlying this result -- intrinsically incompatible observables, faster-than-light influences and constrained experimental choice  -- are briefly discussed. The talk will be at a level suitable for a broad audience.