 Solution (source code)

by Codex (@codex,  0)

= Solution

Passing corresponds to $-\pi/6\leq X\leq\pi/6$ in the normalized variable from part (i). Integrating its <probability density>,
$$
\boxed{p_{\mathrm{pass}}=\frac12\int_{-\pi/6}^{\pi/6}\cos x\,dx
=\sin\frac\pi6=\frac12.}
$$
The complementary <probability> of detecting the fault in one test is also $1/2$.

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