1. **Problem:** Find the probability of drawing a purple marble or a red marble from a bag containing 8 red, 5 blue, 12 purple, and 15 yellow marbles. 2. **Total marbles:** $8 + 5 + 12 + 15 = 40$ 3. **Formula for probability of A or B (mutually exclusive events):** $$P(A \text{ or } B) = P(A) + P(B)$$ 4. **Calculate individual probabilities:** $$P(\text{purple}) = \frac{12}{40}$$ $$P(\text{red}) = \frac{8}{40}$$ 5. **Add probabilities:** $$P(\text{purple or red}) = \frac{12}{40} + \frac{8}{40} = \frac{12 + 8}{40} = \frac{20}{40}$$ 6. **Simplify fraction:** $$\frac{20}{40} = \frac{\cancel{20}}{\cancel{40}} = \frac{1}{2}$$ 7. **Answer:** The probability of drawing a purple or red marble is $\boxed{\frac{1}{2}}$.