Spin 1/2 Particle: Uncertainty Principle Explained

[Molar]

In my book (Griffiths) it is said :

A particle is in ↑ state.
The z-component of the particle's spin angular momentum [S_z] is h bar/2.
The x- component of the particle's spin angular momentum [S_x] can be either h bar/2 or
-h bar /2 .

He says this is for uncertainty principle...that both S_z and S_x can't have definite value.
This is where i am having difficulties.

I mean it is always strictly like this that z component would give certain values and uncertainty would follow
x-component...?
or just because we have measured z component first...ie, if we had measured x-component first then we would
have the value h bar /2 for x- component and the uncertainty principle would act on z-component...??
or because spin is always defined along Z axis...?

thanks...

 

[Doc Al]

or just because we have measured z component first...ie, if we had measured x-component first then we would
have the value h bar /2 for x- component and the uncertainty principle would act on z-component...??

Right.

The particle had to have been prepared in the ↑ state with respect to the z-axis. It could have been any axis--they just like to pick on the z-axis for some reason.