1. **State the problem:** We have 8 balls numbered 1 to 8. Balls 1, 2, and 6 are grey (not white), and balls 3, 4, 5, 7, and 8 are white. We want to find the probabilities of selecting a white ball (event $X$) and not selecting a white ball (event $\text{not } X$). Then, we will verify the relationship between $P(X)$ and $1 - P(X)$. 2. **Formula and rules:** Probability of an event $E$ is given by $$P(E) = \frac{\text{number of favorable outcomes}}{\text{total number of outcomes}}$$ The complement rule states: $$P(\text{not } E) = 1 - P(E)$$ 3. **Calculate $P(X)$:** Number of white balls = 5 (balls 3, 4, 5, 7, 8) Total balls = 8 $$P(X) = \frac{5}{8}$$ 4. **Calculate $P(\text{not } X)$:** Number of non-white balls = 3 (balls 1, 2, 6) $$P(\text{not } X) = \frac{3}{8}$$ 5. **Check complement rule:** $$1 - P(X) = 1 - \frac{5}{8} = \frac{8}{8} - \frac{5}{8} = \frac{3}{8}$$ This matches $P(\text{not } X)$. 6. **Answer for (c):** $1 - P(X)$ is the same as $P(\text{not } X)$. **Final answers:** - $P(X) = \frac{5}{8}$ - $P(\text{not } X) = \frac{3}{8}$ - $1 - P(X) = \frac{3}{8}$ - $1 - P(X)$ is the same as $P(\text{not } X)$