But, my question is whether there is a equation describing which discrete energy states electrons will be in en route to their ground state? Or, is it completely random?
We are obviously in the quantum mechanical regime, which means that any calculations have to obey the postulates of quantum mechanics and the functions found will be describing probabilities and not specific measurements.
The Bohr model is an approximation to the quantum mechanical solution, as the electrons are in an orbital, and not an orbit.
The little dots are single measurements, and only the probability of an electron making a transition can be calculated, so there is no formula for a single measurement, but for an accumulation of measurements.
What levels have high probability for an excited electron to start cascading to the ground state, will depend on the mathematical solutions for the problem. If there is an overlap of orbitals ( in the picture they are all for one energy level, n quantum number) the probability will be higher. That's the way quantum mecanics calculations go.
(here is a complicated experiment using cascades, as an example of mathematical calculations)