We know quantum mechanics gives a random result when we observe a particle that's in a superposition, but why is it random? One of the explanations I've heard is that because light comes with those discrete energy packets called photons, when a photon is passing through a polarized filter, it must either all pass through or all be blocked. You can't let a fraction of the photon pass through while others are blocked. Is it correct? It seems reasonable, but I couldn't find any proper source about this statement.
If it helps, it's not that the nature of the universe is random, it's that we model it as random in Quantum Mechanics.
There are many cases in science where we cannot model the actual behavior of a system, due to all sorts of effects like measurement errors or chaotic behaviors. However, in many cases, we don't need to care about exactly how a system behaves. We only need to worry about the statistical behavior of the system.
Consider this. We are going to roll a die. If it lands 1, 2, or 3, I give you \$1. If it lands 4, 5, or 6, you give me \$1. It is theoretically very difficult for you to predict whether any one roll is going to result in you giving me \$1 or me giving you \$1. However, if we roll this die 100 times, we can start to talk about expectations. We can start to talk about whether this die is a fair die, or if I have a weighted die. We can model the behavior of this die using statistics.
We can do this until it becomes useful to know more. There are famous stories of people making money on roulette using computers to predict where the ball is expected to stop. We take some of the randomness out of the model, replacing it with knowledge about the system.
Quantum Mechanics asserts that the fundamental behavior of the world is random, and we back that up with statistical studies showing that it's impossible to distinguish the behavior of the universe from random.
That's not to say the universe is random. There may be some hidden logic to it all and we find that it was deterministic after all. However, after decades of experimentation, we're quite confident in a whole slew of ways the universe can't be deterministic. We've put together experiment after experiment, like the quantum eraser, for which nobody has been able to predict the behavior of the experiment better than the randomness of QM.
Indeed, the ways the universe can be deterministic are so extraordinary that we choose to believe the universe cannot be that fantastic. For example, there's plenty of ways for the universe to be deterministic as long as some specific information can travel instantaneously (faster than light). As we have not observed any way to transfer information faster than light in a normal sense, we are hesitant to accept these deterministic descriptions of quantum behavior (like the Pilot Wave interpretation).
And in the end, this is all science ever does. It can never tell us that something is truly random. It can never tell us what something truly is. What it tells us is that the observed behaviors of the system can be indistinguishable from those of the scientific models, and many of those models have random variables in them.
As Feynman said when laying out the first principles of quantum mechanics:
How does it work? What is the machinery behind the law?” No one has found any machinery behind the law. No one can “explain” any more than we have just “explained.” No one will give you any deeper representation of the situation. We have no ideas about a more basic mechanism from which these results can be deduced.
We do not know how to predict what would happen in a given circumstance, and we believe now that it is impossible—that the only thing that can be predicted is the probability of different events. It must be recognized that this is a retrenchment in our earlier ideal of understanding nature. It may be a backward step, but no one has seen a way to avoid it.
That statement in bold re probability is what @SuperCiocia is saying.
It's weirder than you thought.
The wavefunction itself is fully deterministic. People often say "it's the measurements that are probabilisitic" but that isn't right either. The measurement is deterministic if you include the measurement apparatus in the wavefunction. And therein is the core of the great mystery, and the big philosophical questions of whether we should include ourselves in the wavefuncion. Mathematically speaking, we should, and that gives us the Many Worlds interpretation.
The real question is: why do I subjectively experience a probabilitic outcome? We don't have the pholosophical answers to what "I" and "experience" refer to in that sentence. Another way to put it is that the real question is why don't I experience the whole of the wavefunction?
If a conscious mind can (for reasons unknown) only experience one outcome of the many which all truly do actually happen then a probabilistic subjective experience may be the only possible experience. It then raises the question of how we associate probabilities with the wavefunction. Why is the probability proportional to the square of the amplitude? No one really knows, but perhaps there is a deep explanation hinted at here although I confess I do not fully understand it myself, but again the answer may be it's a mathematical necessity.
a) I wouldn't call it "random" but "probabilistic".
b) The evolution of a system is fully deterministic. It's the outcome of measurements that is probabilistic.
c) Your reasoning is wrong. The probabilistic nature of measurements' outcomes is something intrinsic to quantum mechanics (the measurement problem), independent of the specifics of the measurement apparatus.
You are asking why QM is random (which in your case given the context is used as probabilistic), and what is correct to say is that QM is probabilistic in nature, and our underlying world, and our universe seems to us to be quantum mechanical, and truly probabilistic.
is there a way to understand the system as having an initial state which forced it to come to this conclusion," the answer is a qualified "no": there are hidden-variable interpretations like the pilot-wave theory which interpret quantum mechanics as a deterministic theory containing unknowable global information.
The point is global. There are quantum effects that cannot be understood in classical terms.
using some thought experiments (my favorite is a game called Betrayal) one can prove that there are quantum effects which cannot be understood in terms of classical local information
Now the universe ultimately is quantum mechanical, and probabilistic. There might be some underlying mechanism, that is not understood by us, but some specifically state that this underlying mechanism, that would make the universe seem fully deterministic to us, cannot be known. The error is not in our measuring devices, we know that we cannot learn about the underlying mechanism (even if there is one).
In a deeper sense randomness is our way of reasoning about information that we do not know, whether there is some unknowable information which makes everything deterministic, it is known that we cannot (not just do not) know it.
So the answer to your question is, that the error is not in our measuring devices, the universe looks to us truly probabilistic, and QM is the best way to describe it that best fits the experiments. QM is simply probabilistic because it describes (models) a universe that appears to us to be truly probabilistic in nature, and there is no (to our knowledge) underlying (more fundamental) mechanism.
Quantum Indeterminacy is Key to the Arrow of Time
There is no machinery to explain the randomness (as Mr. Anderson answered from Feynman), but maybe a connection to other phenomena can help.
I'm going to go out on a limb here, because answers in this forum are supposed to be from established science. But I think I can make a case for an important explanation that I think follows logically, even if I haven't seen in the literature.
Special Relativity and Time Reversal
We know from Special Relativity that all inertial frames are equally valid, that the laws of physics in one (non-accelerating) frame are exactly the same as in any other. This principle also applies to frames of reference where time is reversed. In fact the Feynman-Stueckelberg interpretation of antimatter is the idea that antimatter is matter going backwards in time.
Time Reversal and Entropy
But we know from the second law of thermodynamics that entropy either increases or stays the same, but it doesn't decrease (at least not on the macro scale). So one principle says that the laws of physics are the same under time reversal (actually something called CPT) but another says that entropy increases are irreversible.
This contradiction is called Loschmidt's Paradox.
Time Reversal and Quantum Choices
Now here's the idea that I came up with. It's probably already out there somewhere, I've looked and haven't seen it though. If someone knows where this has been developed (if it has) I would very much like a reference.
If a sequence of events is deterministic (one with no random quantum choices) then the time reversal of that sequence must also be deterministic, and the reversal of that sequence would always return the system to its original state.
But if a sequence of events involves random quantum choices, then the reversal of that sequence also involves random quantum choices, and those choices don't have to return the system to its original state when time is rolled back to the original time.
A photon goes towards an atom, its absorbed by that atom, the atom waits a random amount of time, then it emits a photon in a random direction, and the photon moves away from that atom.
If we could start with the end of this sequence and reverse time, then we get the same kind of sequence, but the time the atom exists in an excited state doesn't depend on the original time and so is probably not going to be the same amount of time, and the direction the photon is emitted is also random, so is probably not going to be in the original direction.
So we can have both the rules of physics be the same between a frame going forwards in time and backwards in time, and still have the forwards and reversed sequences be different, as long as there are random quantum choices in that sequence.
So I think the resolution to Loschmidt's Paradox is this: If entropy increases in a process and so the process is irreversible, it must involve random quantum choices. If a process is deterministic, and doesn't involve random quantum choices, then it must also be reversible and so the entropy in that system will stay the same.
We don't even know that the universe is fundamentally random. That's just the most popular interpretation (called the Copenhagen Interpretation). In this interpretation, the behavior of particles is probabilistic with no deeper reasoning, and the "why" is left to philosophers (or, possibly, a future Theory of Everything).
There are other interpretations in which the universe is not fundamentally random. Hidden variable interpretations say that QM is actually deterministic, but we deal with probabilities due to not having enough information about some hidden variables.
This seems like the most logical first guess. However, due to Bell's Theorem discovered in the 60's, we know that any deterministic QM interpretation must necessarily be non-local - that is, it requires all particles in the universe to be somehow connected to one-another, and able to communicate at faster-than-light speed.
So basically, physicists are more willing to discard determinism than discard locality.
Quantum mechanics is random or, more accurately, probabilistic, because nature is fundamentally not deterministic. Of course there are those clinging to deterministic explanations, like Bohmian mechanics, by ignoring mathematical proofs, just as there are those clinging to Dingle's argument against relativity. But the argument "I don't understand the proof, therefore the proof is wrong" is not a valid scientific argument, even if the arguments disproving determinism are considerably harder to understand than the arguments proving that Dingle was wrong.
The Schrodinger equation may well appear deterministic, but it only determines probabilities; probabilities do not determine results. Quantum probabilities obey a different mathematical structure from classical probability theory precisely because classical probabilities are determined by unknowns or "hidden variables". The mathematical structure of quantum mechanics is as it is precisely because there are no hidden variables determining measurement results.
There are numerous mathematical proofs of this fact, starting with von Neuman (1936). Further proofs have been given by Jauch & Piron (1963), and by Gudder (1968), and many others, but they are sufficiently abstract that few physicists understand them. Kochen and Specker gave a proof which more physicist understand in 1967. Bell himself gave a proof in 1966 (but written earlier), based on work by Gleason, only Bell still didn’t understand the proof, and claimed there was something wrong with it. Bell himself gave a proof in Bell's theorem (1964), which has been generally accepted because it is directly testable in experiment, and is less abstract than other proofs, requiring only that classical probability theory is refuted by experimental evidence, which has since been obtained.
I have given deeper discussion in my second books, and two demonstrations that nature is fundamentally not deterministic in my third (see my profile for links)