Question: Could someone please explain this [experiment](https://en.wikipedia.org/wiki/Stern%E2%80%93Gerlach_experiment#Sequential_experiments) to me? Why depending on the orientation of the apparatuses, we get different results? For example why if we have $z,x,z$ (I am not sure what is the correct notation for this but I hope it is clear what I mean by this) orientation, we get 50-50 possibility to observe the particle in $+z$ and $-z$, while it would change to 100-0 if we remove the apparatus that is parallel to the $x-$axis. I hope this question is clear enough to be answered if not please let me know to clarify it.

Answer: 
I will add an informal answer to the already existing one. The z-component of the spin of an electron and the x-component of the spin of an electron are non-commuting observable (i.e. $\hat{S\_z}\hat{S\_x} \neq \hat{S\_x}\hat{S\_z}$) in quantum mechanics. This implies that you cannot both know the x-component of the spin and the z-component of the spin of a particle at the same time. The punchline is that the very act of measuring your particle changes its state. Hence even if you knew that the particle was spin-up in the z-direction initially, if you measure the spin component in the x-direction afterwards, then you don't know anymore what the spin in the z-direction is. The act of measuring the x-component of the spin changed the state of your electron.
Stern-Gerlach experiments can be modeled as projection operators. This works nicely in spin-1/2.

The projection operator for the +z experiment is $(1+\sigma\_z)/2$ while the -z experiment is $(1-\sigma\_z)/2$ and +x is $(1+\sigma\_x)/2$.

Consecutive experiments are modeled by products of the projection operators (starting on the right). So the +z, +x, -z experiment is modeled by:
\begin{equation}
(1-\sigma\_z)/2\;\;(1+\sigma\_x)/2\;\;(1+\sigma\_z)/2 = \frac{1}{2}
\left(\begin{array}{cc}
0&0\\1&0
\end{array}\right)
\end{equation}
so the action of this series of experiments is to (a) decrease the amplitude to a factor of 1/2 and (b) take spin-up and make it spin-down. In particular, the amplitude need not be zero.

For the sequence +z and then -z, the product is zero:
\begin{equation}
(1-\sigma\_z)/2\;\;(1+\sigma\_z)/2 = 0
\end{equation}
Since the product is zero, no particles get through.

Note that the zero product is because the projection operators "annihilate" each other while putting +x in the middle allows the state to be modified slowly. By making a series of infinitesimal experiments you can arrange for the losses to approach zero.