Square root of a probability

Question

It's well known that if in quantum mechanics the quantity

$$ J=\int_0^{\infty}dx|\Psi(x)|^2 $$

satisfies $$J=1$$ the $$|\Psi(x)|^2$$ represents the probability to find a particle in $x$.

In Hellinger distance we have the following formula: $$H^2(P,Q)=\int d\lambda\left(\sqrt{\dfrac{dP}{d\lambda}}-\sqrt{\dfrac{dQ}{d\lambda}}\right)^2,$$ where $P$ and $Q$ are probabilities. The question is:

What is, if any, the physical meaning of the square root of a probability?

Answer 1

This may not relate to quantum physics, but sometimes the square root of a probability isolates the variable of interest when it's multiplied with itself to produce side effects.

Take dominant and recessive alleles, for example. Let the dominant allele frequency is $p$ and the recessive frequency is $1-p$, or $q$. If the dominant phenotype (observed effect) is $0.51$, it is not immediately useful, as we have $0.51=p^2+2pq$. Rather, it is more useful to take the square root of $q^2=0.49$ to get $q=0.7$, as this is a much more meaningful, indicative variable.

Answer 2

The concept comes from considering particles as waves. Waveforms have components which are amplitude and phase. Moreover, experiments show what is measured to detect a particle is its amplitude ( although its phase can be detected ) as a probability distribution ( double-slite experiment with a source of the electron ). As a result, what satisfies mathematical formulation ( dot product in Hilbert space of random variables ) and coincides with experiments is the square root of probability as the amplitude of the waveform of the particle.