Single qubit gate and measurement $[20$ points$]$

What is the output of the following quantum circuit? Recall $H$ is the Hadamard gate.

$\frac{1}{\sqrt[\phantom{2}]{3}}|0$:

If we measure the state output from this circuit in the computational basis, with what

probability do we get the outcomes $|0$: and $|1$: $?$

$\frac{1}{\sqrt[\phantom{2}]{3}}|0$: