Entanglement symmetry, amplitudes, and probabilities: inverting Born's rule

Symmetry of entangled states under a swap of outcomes ("envariance") implies their equiprobability and leads to Born's rule pk =| ψk|(2). Here I show the converse: I demonstrate that the amplitude of a state given by a superposition of sequences of events that share the same total count (e.g., n detections of 0 and m of 1 in a spin-1/2 measurement) is proportional to the square root of the fraction-square root of the relative frequency-of all the equiprobable sequences of 0's and 1's with that n and m.