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.

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