In Newtonian Mechanics there is no randomness involved once you know all the initial data. In fact, let $M$ be the phase space of a classical system. The points of $M$ are pairs $(q,p)$ of coordinates and momenta.
The time evolution of the system is described by ordinary differential equations on $M$, and once you know the initial conditions, by the existence and uniqueness theorem for solutions of ODE's you get the result that there is a single path in phase space corresponding to the sequence of states parametrized by time, in other words, a single association
$$t\mapsto (q(t),p(t)).$$
The trouble is that if the system has a huge number of particles the problem becomes extremely hard to tackle. Because of this one studies systems like this with Statistical Mechanics and then start dealing with means and so forth. But if you knew all the initial data and could solve the equations of Classical Mechanics (existence is guaranteed, but it might be very hard as I said), there is a unique path of evolution with no randomness whatsoever.
EDIT: Let's tackle this from a different point of view. In Quantum Mechanics a system is described by a state space $\mathcal{H}$, which mathematically is a Hilbert space. The elements of $\mathcal{H}$ are vectors called state vectors which we denoted like $|\varphi\rangle$. It turns out that if you know the state of the system, meaning that you know what state vector describes it, you still don't have full information about the system.
One example: consider a single particle with spin $1/2$. The spin can be either up or down. If the spin of the particle is up, the state of the system is $|\uparrow\rangle$ and if it is down the state is $|\downarrow\rangle$. These states are simple to understand, but that's not all. The most general state is $|\varphi\rangle = a |\uparrow\rangle + b |\downarrow\rangle$ and in this state everything you can say is that there is a probability of $|a|^2$ that when the spin is measured it will be up and $|b|^2$ that when the spin is measured it will be down.
And that is not all. Even in the state $|\uparrow\rangle$ you can't know the $x$ and $y$ components of spin. You just know the $z$ component is $1/2$. All you can get are probabilities.
So in QM even if you know the state, you don't know it all. There is randomness that is part of the theory.
Nevertheless the evolution is deterministic. Given one initial state, there is precisely a single evolution. But that's not the point. The system will evolve to some other state like these I've examplified, and in the state there will be inacessible information about the system. Again, deterministic evolution guarantees that you can evolve an initial state in a unique manner, but even if you know the state, you can't know it all.
Classical Mechanics isn't like that. In a Classical System you can know both position and momentum exactly in each state. Every observable is a function of position and momentum, hence you know any physical quantity if you know the state. Together with the fact that the evolution in time is unique, if you know the initial state you know it all.
Again you need to know exactly: (i) the initial conditions, (ii) the solution to the equations of motion. It is guaranteed to exist, not guaranteed to be easy to know.