Question: I have an example that I don't understand, especially 2 expressions.

Given a spin in the state $|z + \rangle$, i.e., pointing up along the z-axis what are the
probabilities of measuring $\pm \hbar/2$ along $\hat{n}$?

I have these 2 expressions

$$|+\rangle\_n = cos(\theta/2)e^{-i\phi/2}|+\rangle +sin(\theta/2)e^{i\phi/2}|-\rangle$$
$$|-\rangle\_n = sin(\theta/2)e^{-i\phi/2}|+\rangle -cos(\theta/2)e^{i\phi/2}|-\rangle$$

I'm not sure to fully understand what that means.
Is $|+\rangle\_n$ only used for a particle in the spin $|z + \rangle$ and $|+\rangle\_n$ is a linear combination of $\hbar/2$ and $- \hbar/2$.
Hence, the probability to have $\hbar/2$ is $|cos(\theta/2)e^{-i\phi/2}(1,0)|^2$, knowing that $|+\rangle = (1,0)$

I'm not that convince about my explanation. This is not based on anything.

Answer: 
> 
> If I'm sitting on the edge of a rotating merry-go-round, then from my perspective the world is rotating around me.
> 
> 
> 

No, it isn't. Think about this carefully (or find a merry-go-round and do a practical experiment). From your perspective the merry-go-round is stationary and the rest of the world is rotating *about the centre of the merry-go-round*.

Think about the point in the outside world that is nearest to you. When the merry-go-round has rotated through $180^o$ this point is now on the opposite side of you *and the opposite side of the merry-go-round*. If the world appeared to be rotating about you then this point would stay the same distance away from you as the merry-go-round turned, which it clearly does not.

Therefore, whether you think of yourself as stationary and the world as rotating or vice versa, there is only one fixed centre of rotation, which is always the centre of the merry-go-round.
If there is a non rotating object (like a pole) in the center of the merry go round, this object will look rotating around itself, from the edge person perspective.

But all other non rotating objects (the rest of the world) will rotate around the pole in the center.

It is easy to see that for non rotating objects close to the person in the edge at some moment. Of course those object will be far away after half a cycle. So they are not rotating around the person.