There are completely random events in the realm of quantum physics. The notion that enables this is called "incompatible observables". If you know one, you can impossible know the other (the
Heisenberg uncertainty principle). One classic example of incompatible observables is position and momentum.
A better example is the following: There is a property called "spin", the details of what it is are kinda complicated, but it can sort of be likened to the rotation of an electron or a proton or some such. If you know the spin in one direction (say vertically), you can not know the spin horizontally. In fact, if you know the spin vertically, and then measure the spin horizontally, and then re-measure the spin vertically, you will get a completely random value. This can be shown experimentally, and is called the
Stern-Gerlach experiment.
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