I was about to say that, but I wasn't sure that it's true. From probabilities you can compute averages, but I'm not sure about the other way around. For example, suppose that I tell you that [itex]\langle S_z \rangle = 0[/itex]. That doesn't give me much information about probabilities of various values of [itex]S_z[/itex].In physics, probabilities and averages give the same information, so I accept vanhees71's definition of the Born rule.
But maybe if for some observable [itex]A[/itex], I know [itex]\langle A^n \rangle[/itex] for every [itex]n[/itex], does that uniquely determine the probability?