1. **State the problem:** We have a dartboard with 5 equal slices numbered 1 to 5. - Slices 2, 3, and 4 are grey. - Slices 1 and 5 are white. A dart lands randomly on one slice. We define event $X$ as landing on a grey slice, and event $\text{not } X$ as landing on a non-grey slice. 2. **Calculate probabilities:** Since all slices are equally likely, $$P(X) = \frac{\text{number of grey slices}}{\text{total slices}} = \frac{3}{5}$$ $$P(\text{not } X) = \frac{\text{number of non-grey slices}}{\text{total slices}} = \frac{2}{5}$$ 3. **Check outcomes for each event:** - $X$ contains slices $\{2,3,4\}$. - $\text{not } X$ contains slices $\{1,5\}$. 4. **Subtract:** $$1 - P(X) = 1 - \frac{3}{5}$$ Show cancellation: $$1 - \frac{3}{5} = \frac{5}{5} - \frac{3}{5} = \frac{\cancel{5}}{\cancel{5}} - \frac{3}{5} = \frac{2}{5}$$ 5. **Interpretation:** $$1 - P(X) = P(\text{not } X)$$ This means the probability of not landing on a grey slice is the complement of landing on a grey slice. **Final answers:** - $P(X) = \frac{3}{5}$ - $P(\text{not } X) = \frac{2}{5}$ - $1 - P(X) = \frac{2}{5}$ - $1 - P(X)$ is the same as $P(\text{not } X)$.