Born rule

You are given an experimental set-up with an electron that can either occupy a quantum state or not occupy it. Which state represents the electron if you know the probability of occupying the state is 0.25?

Considering that the state has to be properly normalized and that there is an arbitrary phase between the occupied state, which we define as |1>, and the not-occupied state, which we define as |0>, the answer is:

$|\psi\rangle=\sqrt{0.75}|0\rangle+e^{i\phi}\sqrt{0.25}|1\rangle$

What is the state of the electron if now you know the probability of occupying the state is 0.1?

Similarly, the answer in this case is:

$|\psi\rangle=\sqrt{0.9}|0\rangle+e^{i\phi}\sqrt{0.1}|1\rangle$

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Born rule