It depends on your abstraction level. But if you want a single answer...
Probabilistically determined...at least that is my interpretation of currently accepted science...and I am not including string theory since it hasn't had much (if any) empirical testing.
There are 4 fundamental interactions. There are two forces of limited scale (that is distances that it effects stuff)...the strong nuclear force (responsible for holding nuclei together), the weak nuclear force (responsible for beta decay and other such things). The two other forces have infinite scale, and they are the more familiar electromagnetic force (electric field and magnetic field depends on your reference frame) and the force of gravity.
At base, the universe of particles is governed by Schroedinger's Equation (or rather Dirac's Equation when you account for special relativity), which is an equation of the governing the square root of the probability that particle will be measured at a particular position. The equation also happens to be a wave equation.
Similar equations, Quantum Electrodynamics, describe electromagnetic forces.
Quantum chromodynamics, describes strong nuclear forces.
A theory of weak interactions mimics the theory of electrodynamics except with different mediators.
These theories are all probabilistic in nature. The descriptions are neither complete randomness, not completely determined.
Then there is the completely deterministic description of gravity known as General Relativity.
So at the physical microscopic level things are probabilistic.
But as you get more macroscopic, the averaging effects (law of large numbers, central limit theorem, statistical mechanics, etc.) make the physical world LOOK completely deterministic. This is the world described by classical mechanics and classical electrodynamics.
But as you keep going up in abstraction..chemistry, biology, etc. Things start getting very dynamic and hard to predict, but there remains regularity and an inkling that, underneath it all, it is deterministic. Nevertheless, we model things with a mix of completely deterministic and probabilistically determined ways.
Then we get to the level of individual people...here things are extremely dynamics and essentially chaotic. Psychology, Philosophy, History, etc. all attempt to describe this level, but we are left with things like "consciousness," "choice," and other phenomenon that we are hard pressed to even explain probabilistically. Are these phenomena due solely to complexity, or does the base fundamentally probabilistic nature also play a role?
Then we go up in abstraction and we start to study people in aggregate. This is the realm of demographics, sociology, economics, etc. Here again, we are aided by averaging effects. Things become more predictable, and somewhat amenable to probabilistic descriptions. However, the theories often break down, due to things referred to as "Black Swans" (rare events that have profound effects).
I applaud those who read this far. This is something, I think about a lot.