Do excited electrons (by radiating) get into lower energy state(higher than ground state) preferentially or randomly en route to the ground state? Electrons in atom on lower energy states can absorb energy to get into higher energy states. Usually, electrons get to their ground state after reradiating energy.
But, electrons do not always get back at once to the same lower energy level that they used to be in. They can get into some discrete lower energy states(higher than the ground state though) en route to the ground state.
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?
 A: The energy levels are quite clearly defined. For instance, atomic hydrogen is divided into a number of spectral series, with wavelengths given by the Rydberg formula.
What determines how many levels an electron may transition and the means by which the energy is released or dissipated during relaxation (transitioning to a lower energy state) is not a straightforward process but has a number of influences. 
The basic process of photon absorption/emission is explained fairly well at SDSS - Energy Levels of Electrons - essentially a photon with energy that matches an electron transition energy requirement can excite an electron to a higher state, and when the electron drops back a photon is emitted with the energy of the transition.
A more detailed explanation about Jablonski diagrams and the different mechanisms can be found at Chemistry Libretexts - Jablonski diagram, which describes not just the processes of excitation and relaxation but also describes typical timeframes that the different transitions take. 
But the time that an electron may stay in some intermediate level before dropping back to the ground state is going into the realm of quantum physics and is determined on more of a probability basis rather than having a timeframe that can be calculated by means of a formula.
A: 
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)
