I have the general state of a single qubit $|\psi \rangle = \alpha|0\rangle + \beta|1\rangle $. Assume I am given a probability $p$ such that $0 < p <1$. Now I need to find the basis in which the measurement of the qubit $|\psi \rangle $ gives outcome $0$ with probability $p$.
I understand that the required basis $B$ will have $2$ states, so assume the basis is $B =\{|a\rangle, |b\rangle \}$. Also that, probability of outcome $0$, when measured in basis $B$, is $|\langle a |\psi\rangle|^2 $ which should equal $p$. At the same time, probability of outcome $1$ in basis $B$ is $|\langle b |\psi\rangle|^2 $ and should equal $1-p$.
I do not understand how to solve the above two inner-product equations to obtain states $|a\rangle$ and $|b\rangle$?