So I've just been reading about the Stern-Gerlach apparatus/experiment and had some questions about the half spin probabilities... First let's say a collection of spin-half particles are sent through a S-G apparatus (which I know measures the z component of particles spin angular momentum) it is known that the results are always either 1/2ħ or -1/2ħ. And though the result cannot be predicted before it has been calculated by quantum mechanics that the probability of getting 1/2ħ is 0.75. So then what is the probability of getting the result -1/2ħ? I saw some sites referring to cosine and sine while others said it was 1 so it confused me So for example if say 20000 particles are sent through, how many will have a z component equal to -1/2ħ (I'm thinking 50%?) then what would be the average z component of spin for all the particles?