The core of the theory is an algebra of observables. These include position coordiates, momentum, energy, spin, electric charge, and anything else that is measurable as a real variable.
The key fact is that the observables do not commute. That is, the position X and the momentum P have the property that XP is not equal to PX. They differ by h-bar, a small quantity called Planck's constant.
This is not so radical, as many everyday observables also have this property that observations depend on the order that they are performed. For example, poll questions:
Sometimes the very order of the questions can have an impact on the results. Often that impact is intentional; sometimes it is not. The impact of order can often be subtle.For more, see Why Question Order Changes Poll Results.
During troubled economic times, for example, if people are asked what they think of the economy before they are asked their opinion of the president, the presidential popularity rating will probably be lower than if you had reversed the order of the questions. And in good economic times, the opposite is true.
To observe a system, we need a representation of the observables on a Hilbert space of possible system states. That means that a vector ψ represents the state of the system, that an observable A acts on ψ to give a new state Aψ, and that two vectors ψ1 and ψ2 can be combined to get a number <ψ1|ψ2>. The latter is like an ordinary dot product and gives 0 when the vectors are orthogonal.
If an observable A is measured is measured on a system state ψ, the expected value is <ψ|Aψ>, also written <ψ|A|ψ>. It is a real number.
An actual lab measured value may not match the expected value exactly. Real numbers never match exactly in the lab, with quantum mechanics or any other scientific theory. The standard deviation, or sigma, is also an observable with an expected value. Thus, the mechanics might say that a particle will be observed at a distance of 5.24 ± .03 meters. Then a measurement is likely to be between 5.21 and 5.27.
So quantum mechanics make probabilistic predictions in the sense that it gives a range of likely outcomes for measurements. But every other branch of science does something similar, and this is not why quantum mechanics is said to be probabilistic.
Quantum mechanics is said to be probabilistic because is predicts probabilities. Here is how. Suppose that the observable A is a yes-no (boolean) observable, such as asking whether an electron is in a particular region of space. Yes means 1, no means 0, and no other values are observed. Then the expected value <ψ|A|ψ> will be in the range [0,1]. If the value is 1, then you can be sure of a yes, and if the value is 0, then you can be sure of a no. If the value is in between, then it can be interpreted as a probability of a yes. This interpretation is called the Born rule. Max Born suggested it as one possibility in a 1926 paper footnote, and got a Nobel prize for it in 1954.
Probabilities do not play an essential role here. Testing the Born rule is just a special case of testing an expected value of an observable, where the observable is a yes-no variable. An experiment does not really say whether there is any genuine randomness. It just says that if the expected value of a yes-no observable is 0.65, and you do 100 experiments, then you should get about 65 yes outcomes.
It is better to just say that quantum mechanics predicts the expected values of observables. That is what the formulas really do, and that is how the theory is tested. The Born rule adds an interpretation in the case of a yes-no observable. But that interpretation is just metaphysical fluff. There is no experimental test for it. The tests are just for the expected values, and not for the probabilities.
Thus I do not believe that it is either necessary or very useful to talk about probabilities in quantum mechanics. I guess you could say that the probability gives a way of understanding that the same experiment does not give the same outcome every time, but it does not give any more quantitatively useful info. This understanding is nothing special because every other branch of science also has variation in experimental outcomes.
My view here is a minority view. I have not seen anyone else express it this way. The textbooks usually say that ψ is a probability density or amplitude. But ψ is complex-valued or maybe even spinor-valued, and it requires some computation to get a probability. It is not a probability. That computation is precisely the expectation value described above. Sometimes the textbooks admit that the quantum probabilities require special interpretation because they can be negative. I say that negative probabilities are not probabilities and that the probabilities are no more essential to quantum mechanics than to any other physical theory that does real number computations.
I reader asks what quantum interpretation this is. It is similar to the ensemble interpretation, without the probabilities.