Solution
= Solution
Conditional on $\Phi=\phi$, the <uniform distribution> of the sector's centre angle gives a range of possible centre bearings of length $\phi$ that hit a specified house. Independence of the two angles therefore gives the <conditional probability> $\phi/(2\pi)$. By the <law of total expectation>,
$$
\boxed{\mathbb P(H_1\text{ hit})=\mathbb E\left[\frac{\Phi}{2\pi}\right]=\frac13.}
$$