Well, yes and no. Quantum-mechanically it IS deterministic in the sense that any given quantum state will evolve in a perfectly defined way. There isn't any "random number" in the Schroedinger equation (or its relativistic descendants).
It's really the macro-scale stuff that introduces the randomness. At the quantum scale, things exist perfectly happily in a superposition of two states that we never observe at large scales. The more objects you put together, the harder it is to maintain the superposition, and by the time you get to even microscopic objects it will take one state or the other, but not both. Once it tips slightly in one direction, it cascades, and you end up with something that is entirely X or Y, not (X+Y).
The other half of the wave function is largely a matter of philosophy, not physics. In one sense it's "still there", off in some other utterly inaccessible universe. Or you can say that at some point where you weren't looking the other part just vanished. That's two ways of saying the same thing; the math is the same and the results are the same, regardless.
It's not a question of our inability to measure it. It's simply not there. No advances in physics will make it measurable, not without utterly throwing out everything we know and replacing it with something completely different. Which isn't impossible, but it's purely speculative: physics by "I wanna believe".
Why we end up in "this part" rather than "that part" is, similarly, just idle speculation. I've got my suspicions that if you could, in fact, discuss the wave function of the entire universe you'd say that it could only go one way when you put all of it together, but that's just navel-gazing. It doesn't really matter, since you'll never actually know the wave-function of the universe as a whole. You can only observe a few macro parts of it since you (by definition) are a macro organism, and the total underlying wave function will always be forever shaded from your eyes.