Phase You Can Measure

A single qubit starts in $|0\rangle$ and runs this circuit, then is measured in the computational basis:

|0> ---[ H ]---[ Z ]---[ H ]---( measure )

What is the outcome distribution?

0 with probability $\tfrac{1}{4}$ and 1 with probability $\tfrac{3}{4}$ .

1 with certainty

0 and 1 with probability $\tfrac{1}{2}$ each, because $Z$ changes only a phase and phases are unobservable.

0 with certainty, because $Z$ acts only on $|1\rangle$ and the state began in $|0\rangle$ .