Just so you all know, Clifford Algebra derivations of quantized field theory show why the Born Probabilities are a squared proportion. I'm not sure there's an intuitively satisfying explanation I can give you for why this is that uses words and not math, but here's my best try.

In mathematical systems with maximal algebraic complexity for their given dimensionality, the multiplication of an object by its dual provides an invariant of the system, a quantity which cannot be altered. (And all physical field theories (except gravity, at this time) can be derived in full from the assumption of maximal algebraic complexity for 1 positive dimension and 3 negative dimensions). [Object refers to a mathematical quantity, in the case of the field theories we're concerned with, mostly bivectors].

The quantity describing time evolution then (complex phase amplitudes) must have a corresponding invariant quantity that is the mod squared of the complex phase. This mod squared quantity, being the system invariant whose sum describes 'benchmark' by which one judges relative values, is then the relevant value for evaluating the physical meaning of time evolutions. So the physical reality one would expect to observe in probability distributions is then the mod squared of the underlying quantity (complex phase amplitudes) rather than the quantity itself.

To explain it in a different way, because I suspect the one way is not adequate without an understanding of the math.

Clifford Algebra objects (i.e. the actual constructs the universe works with, as best we can tell) do not in of themselves contain information. In fact, they contain no uniquely identifiable information. All objects can be modified with an arbitrary global phase factor, turning them into any one of an infinite set of objects. As such, actual measurement/observation of an object is impossible. You can't distinguish between the object being A or Ae^ib, because those are literally indistinguishable quantities. The object which could be those quantities lacks sufficient unique information to actually *be* one quantity or the other. So you're shit out of luck when it comes to measuring it. But though an object may not contain unique information, the object's mod squared does (and if this violates your intuition of how information works, may I remind you that your classic-world intuition of information counts for absolutely nothing at the group theory level). This mod squared is the lowest level of reality which contains uniquely identifiable information.

So the lowest level of reality at which you can meaningfully identify time evolution probabilities is going to be described as a square quantity.

Because the math says so.

By the way, we're really, really certain about this math. Unless the universe has additional spatial-temporal dimensions we don't know about (and I kind of doubt that) and only contains partial algebraic complexity in that space (and I really, really doubt that), this is it. There is no possible additional mathematical structure with which one could describe our universe that is not contained within the Cl_13 algebra. There is literally no mathematical way to describe our universe which adequately contains all of the structure we have observed in electromagnetism (and weak force and strong force and Higgs force) which does not imply this mod squared invariant property as a consequence.

Furthermore, even before this mod squared property was understood as a consequence of full algebraic complexity, Emmy Noether had described and rigorously proved this relationship as the eponymous Noether's theorem, confirmed its validity against known theories, and used it to predict future results in field theory. So this notion is pretty well backed up by a century of experimental evidence too.

Tl;DR: We (physicists who work with both differential geometries and quantum field theory and whom find an interest in group theory fundamentals beyond what is needed to do conventional experimental or theory work) have known about why the Born Probabilities are a squared proportion since, oh, probably the 1930s? Right after Dirac first published the Dirac Equation? It's a pretty simple thing to conclude from the observation that quantum amplitudes are a bivector quantity. But you'll still see physics textbooks describe it as a mystery and hear it pondered over philosophically, because propagation of the concept would require a base of people educated in Clifford Algebras to propagate through. And such a cohesive group of people just does not exist.

*0 points [-]