= Solution
Evaluating the fitted <linear discriminant analysis> score gives
$$
\begin{array}{c|r|l}
\text{individual}&q(x)&\text{assignment with equal priors}\\\hline
1&-3/4&\text{apathus}\\
2&7/4&\text{fattus}\\
3&-5/4&\text{apathus}\\
4&1/2&\text{fattus}\\
5&0&\text{tie: either species}
\end{array}
$$
\b[The fifth individual lies exactly on the discriminant boundary.] Its two fitted posterior probabilities are equal, so the equal-prior <Bayes classifier> cannot prefer either species. A specified tie convention could assign it to one species, but such an assignment would not be evidence favoring that species.
In the sketch the solid line is the equal-prior boundary. The dashed line is the <prior-dependent Gaussian discriminant boundary> for the altered priors in (iii).
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2006/iii/paper-46-beetles.png]
{title=Beetle classifications and the parallel boundary shift when fattus has twice the prior probability}
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