Determine the state $|\psi \rangle$ The question is:

The angular momentum components of an atom prepared in the state $|\psi\rangle$ are measured and the following experimental probabilities are obtained:
\begin{equation} P(+\hat{z}) = 1/2, P(−\hat{z}) = 1/2,
\end{equation}
\begin{equation}
P(\hat{x}) = 3/4, P(−\hat{x}) = 1/4,
\end{equation}
\begin{equation}
P(+\hat{y}) = 0.067, P(−\hat{y}) = 0.933.
\end{equation}
From this experimental data, determine the state $|\psi \rangle$. Note that in performing the measurements, the state $|\psi \rangle$ is prepared again and again.

My attempt:
$$
 P(+\hat{z}) = 1/2 = P(−\hat{z})
$$
$$
|\langle {\uparrow}_z|\psi\rangle|^2=1/2 =|\langle {\downarrow}_z|\psi\rangle|^2
$$
$$
|\psi \rangle =\alpha |{\uparrow}_z\rangle+ e^{iδb} \beta|{\downarrow}_z\rangle
$$
$$
|\langle {\uparrow}_z|\psi\rangle|= \alpha = 1/\sqrt(2).
$$
Similarly,
$$
|\langle {\downarrow}_z|\psi\rangle|= \beta = 1/\sqrt(2).
$$
However, I don't know how to find $e^{iδb}$ term. Could someone please give a hint?
 A: REFERENCE : My answer here Understanding the Bloch sphere
$=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=$
Equation (24) in my answer above is
\begin{equation}
\vert\psi\rangle \boldsymbol{=}\cos\left(\dfrac{\theta_3}{2}\right)\vert u_3\rangle \boldsymbol{+} e^{i\phi_3}\sin\left(\dfrac{\theta_3}{2}\right)\vert d_3\rangle 
\tag{24}\label{24}
\end{equation}
where $\vert u_3\rangle ,\vert d_3\rangle $ are yours $|{\uparrow}_z\rangle,|{\downarrow}_z\rangle$ respectively.
From the given probabilities $P(+\hat{z}) = 1/2, P(−\hat{z}) = 1/2
$ the state lies on the "equator" of the Bloch sphere. So from Figure-01 in my REFERENCED answer $\theta_3=\pi/2$. The angle $\phi_3=\boldsymbol{-}\pi/3$ could be found from one only of the probabilities  $P(\hat{x}) = 3/4, P(−\hat{x}) = 1/4,P(+\hat{y}) = 0.067$, $P(−\hat{y}) = 0.933$ and Figure-02  in my REFERENCED answer.
Note : I suggest you to "study" the excellent @CR Drost's answer about the Bloch sphere in above link.



See a 3d view of Figure-03 here
