Measuring a qubit in the wrong basis

I'm a computer scientist and right now I'm trying to end the research work for my master thesis and a basic problem of quantum mechanics is blocking me.

I'm trying to do a probability calculus of the action of measuring a qubit in a wrong bases. In bibliography, I've always found the statement:

When Bob chooses the wrong bases for measuring a qubit then the result will be completely random.

But what exactly does that mean? The results will be Non-deterministic (using theoretical CS terms) and then the probability cannot be calculated or that mean that the probability of the result is exactly $$\frac{1}{2}$$0 and $$\frac{1}{2}$$1?

Cross-posted on crypto.SE

• I recommend Feynmann lectures (the quantum mechanics part) for simple but correct explanations. It is essentially what Stern-Gerlach device does (replace qubit by two-level atom/spin and you have it). Nov 26 '21 at 11:42
• My physics background is a little bit poor, however, you mean this? feynmanlectures.caltech.edu/III_10.html Nov 26 '21 at 12:41
• I would say earlier: this one; Spin one-half, but the preceding one can be useful as well, since it explains teh Stern-Gerlach apparatus. Don't be misled by quantum computing buzz-words - qubit is just a two-level system, and spin-1/2 is the simplest one. Physical realization of a TLS may be different, but the math and probabilistic interpretation remain the same. Nov 26 '21 at 12:57
• I assume the "right" basis is the computational basis $|0\rangle, |1\rangle$, what is the wrong basis? You should define the setting and terms. Nov 26 '21 at 14:08