Unfortunately (for me) I am not a physicist. So take this with a grain of salt.
Some thinking about this puzzling issue made me have these conclusions that may be useful for you:
@Quantum world:
1) Entities have a 'spread' existence. (A kind of 'field of matter').
2) Entities have some 'oscilatory' existence. (Which gives rise to 'interference').
3) Interactions between entities are 'discrete'. (They exchange 'quanta' of somestuff).
4) Interactions use the 'minimum amount' of some 'energy stuff'.
The interplaying of these facts is what gives rise to the non-determinism (probability) in QM.
Let's think of a simple example:
Suppose you have 3 entities A, B and C, where A is the source of some perturbation to be sent to B and C at the 'same time'. Let's think of the perturbation in practical terms (i.e.: money) and assign it a unit of measure (dollars).
Now how would A send 2 dollars total to both of them (B & C)?
Well, A should give them 1 dollar each and problem solved!!!. However, there is a constraint here and that is: Interactions are only done with minimun currency!!!'.
With that in mind, how can A give B and C one cent (minimun currency) at the same time? Well, .. It can't!!!
At each time (interaction) A must choose between B or C to give away every cent until completes the 2 dollars to both of them. And if you think a little bit about it, you realize that the only objective solution for A must be to throw an imaginary coin each time to decide whom will receive the 1 cent!.
In the analog world of classical mechanics, A would send an infinite small amount of money to both of them (no minimum currency constraint and at the same time!) and what we will see is a beautiful continuous growing of B and C money pockets. No need to deal with probabilities!!!!.
If you think carefully, in plain simple terms, probability arise from the discrete nature of interactions between entities. This is the real deal which turns everything so strange and interesting.
The question now is: Why it has to be like that? Is there a 'principle of least calculation that Nature follows'???