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It is said that an unobserved electron is at no particular place but has a probability of being at various places.
Is this probability a function of the electron or of our observation?
That is, is there a certain probability of the electron being at x, or is there a certain probability of our making a measurement at x?
Pardon me if this is a naive question.
It is not easy to directly answer, because you are implicitly assuming that the electron is a classical object. So my answer will be necessarily a bit articulate.
The crucial difference between classical and quantum objects is that for a classical object all its properties always have definite values. This requirement is usually called realism. When instead dealing with quantum objects, their properties may have undefined values. These values are however fixed when (after) one measures them (through an interaction between the quantum object and the measurement apparatus).
Quantum theory provides the probability to fix each value of each property after a measurement of it (if I decide to measure it).
A quantum state is nothing but that assignment of all those probabilities for every property and every value of it (this is the physical content of the famous Gleason theorem). When a property has a definite value, its probability is $1$.
There are pairs of quantum properties which are mutually incompatible: they cannot have definite values simultaneously and this explains why there are properties which are not defined in some state. They cannot have definite values because other, incompatible, properties have definite values since I have just measured them. Notice that therefore measurements change the state of the system.
In summary, the position of the electron can be undefined: the electron has literally no position (!). However Quantum Theory gives us the probability that if I measure the position it results to belong to a given interval. If I perform such a measurement, the position localizes, but the state (the probabilities of the outcomes of other measurments) changes and other properties, which were defined before the measurement, become undefined.
One could be tempted to assume some hidden form of realism: the electron always has a position and every property is actually always defined, but, for some reason, we do not know it and we know it only up to a probabilistic estimate (exactly as, e.g., in classical statistical mechanics).
As a consequence of several theoretical investigations (with also several experimental confirmations) generally known as the Kochen-Specker theorem (actually based on previous results by Gleason and Bell), this viewpoint is untenable, unless assuming some weird (strongly non classic) behavior of the "classical" properties called contextuality.
That is, is there a certain probability of the electron being at x, or is there a certain probability of our making a measurement at x?
Both? It certainly depends on the measurement, but your latter statement makes it seem like you are saying QM predicts where I put my detector.
I would say QM predicts the probability of detecting a particle at a certain location (more precisely, within a certain region of space). If we detect the particle to be at some location, we then we can say we "found it there" I suppose. If we don't detect it at some location, we have still "made a measurement", but it's just that we don't observe the particle to be there. But you are correct in saying that before this there is no definite position of the particle.
