Since $P(E) + P(\bar{E}) = 1$, we have $q = 1$.
Therefore, $q^2 - 4 = 1^2 - 4 = 1 - 4 = \mathbf{-3}$.
Answer: (A) $-3$
Source: Chapter 14, Section 14.2 Summary (Point 6)
The key fact is the complementary event rule: $P(E) + P(\bar{E}) = 1$ always. So $q = 1$, and $q^2 - 4 = -3$. Examiners expect you to recall this standard result directly — no working beyond substitution is needed for 1 mark.