Why Probability in Quantum Mechanics is Given by the Wave Function Squared

Sean Carroll in Preposterous Universe:

One of the most profound and mysterious principles in all of physics is the Born Rule, named after Max Born. In quantum mechanics, particles don’t have classical properties like “position” or “momentum”; rather, there is a wave function that assigns a (complex) number, called the “amplitude,” to each possible measurement outcome. The Born Rule is then very simple: it says that the probability of obtaining any possible measurement outcome is equal to the square of the corresponding amplitude. (The wave function is just the set of all the amplitudes.)

Born Rule:

The Born Rule is certainly correct, as far as all of our experimental efforts have been able to discern. But why? Born himself kind of stumbled onto his Rule.

More here.