What is the reason that Quantum Mechanics is random?
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.