= Solution
The <orthogonal projection> onto the $+1$ eigenspace of the <Pauli matrix> $\sigma_z$ is $P_+=(I+\sigma_z)/2$. By the <Born rule> and the <Bloch vector> representation,
$$
\mathbb P(+1)=\operatorname{Tr}(\rho P_+)=\frac12\left(1+\operatorname{Tr}(\rho\sigma_z)\right)=\frac{1+s_z}{2}.
$$
Since $s_z=1/5$,
$$
\boxed{\mathbb P(+1)=\frac35.}
$$
The given vector has length $19/30<1$, so it lies inside the <Bloch ball> and indeed defines a valid mixed qubit state. For the physical spin operator $S_z=(\hbar/2)\sigma_z$, the corresponding outcome is $+\hbar/2$ with the same probability.
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