For a general state vector, does the probability of finding the system in spin up/down change based on direction? I understand that for a general state vector describing a spin half system, the probability of finding the system in a spin up/down state is 50% for both spin up and down, regardless of which direction we are measuring spin in.
However, what if we just have a general state vector with coefficients that aren't 1/sqrt(2)? What if the coefficients of the basis kets are different.
Would the probability of measuring the system to be in Spin up/down in the z direction be different to the probability of finding the system in spin up/down in the x or y direction?
 A: That isn't true.  A general state vector describing a spin-half system can be written
$$|\psi\rangle = c_1|\uparrow\rangle + c_2|\downarrow\rangle$$
where $|c_1|^2+|c_2|^2=1$. The probability of measuring the system to be spin-up is $|c_1|^2$, and the probability of measuring the system to be spin-down is $|c_2|^2$.  These probabilities could be 50/50, but they could be any other combination as well.  It all depends on what the state is.
We have implicitly decided to measure the spin along the $z$-axis, and so we have expressed our state in the $z$-basis.  We could measure the spin along the other axes if we want to - we would simply need a change of basis to find our new probabilities.  Explicitly,
$$|\uparrow\rangle_z = \frac{|\uparrow\rangle_x + |\downarrow\rangle_x}{\sqrt{2}} = \frac{|\uparrow\rangle_y + |\downarrow\rangle_y}{\sqrt{2}}$$
$$|\downarrow\rangle_z = \frac{|\uparrow\rangle_x - |\downarrow\rangle_x}{\sqrt{2}} = \frac{|\uparrow\rangle_y - |\downarrow\rangle_y}{i\sqrt{2}}$$
So if we express our state in the $x$-basis, for example, we would find that
$$|\psi\rangle = \left(\frac{c_1+c_2}{\sqrt{2}}\right)|\uparrow\rangle_x + \left(\frac{c_1-c_2}{\sqrt{2}}\right)|\downarrow\rangle_x$$
So the probability of measuring the system to be spin-down along the x-axis would be 
$$P(\downarrow,x)=\left|\frac{c_1-c_2}{\sqrt{2}}\right|^2 = \frac{1-c_1^*c_2-c_1c_2^*}{2}$$
