Measurement of an entangled Particles in two Different Bases

[randomuser3210]

Consider two entangled spin half particles given by the generic form of Bell Equation in Z-axis:
$\psi = (a\uparrow \uparrow + b\downarrow \downarrow)$ where $a^2+b^2=1$

In a (2D) planer rotated (by an angle $\theta$) direction the new equation can be given by:

$|\psi \rangle = [\alpha \cos^2(\theta/2) + \beta \sin^2(\theta/2)] |\uparrow\uparrow \rangle +$
$[(\alpha \sin^2 \theta/2 + \beta \cos^2 \theta/2)] |\downarrow\downarrow \rangle$
$[(\beta-\alpha) \cos(\theta/2) \sin(\theta/2)] (|\uparrow\uparrow \rangle +|\uparrow\downarrow \rangle +$

Now Alice keeps one particle and sends the other to Bob. Here is the Q:

1. Suppose, Alice measures the particle in $\theta$ direction. Then after that, Bob measures in the Z direction. What is the probability of Bob getting the particle $\uparrow$ and $\downarrow$ in Z directions each.

I know that as soon as Alice measures the particle in $\theta$ direction, the entanglement collapses. So we can measure the probability by using the wave function. But, can someone help with the cases for 'cross measurement'?

 

[mfb]

If you are just interested in Bob's measurement on its own you can ignore what Alice does.
If you need the correlation look at the wave function from the perspective of one of them and calculate the chance that the other one measures one of the outcomes.