1. **State the problem:** Find the probability of rolling a 4 on a six-sided dice and the probability of the complementary event. 2. **Formula and rules:** The probability of an event $E$ is given by: $$P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}$$ The complementary event $E^c$ is the event that $E$ does not occur, and its probability is: $$P(E^c) = 1 - P(E)$$ 3. **Calculate $P(4)$:** There is only one favorable outcome (rolling a 4) and 6 possible outcomes (1 through 6), so: $$P(4) = \frac{1}{6}$$ 4. **Calculate the complementary probability:** $$P(4^c) = 1 - P(4) = 1 - \frac{1}{6} = \frac{6}{6} - \frac{1}{6} = \frac{5}{6}$$ 5. **Answer:** The probability of rolling a 4 is $\frac{1}{6}$ and the probability of the complementary event is $\frac{5}{6}$. --- 1. **State the problem:** Find the probability of rolling a 2 on a six-sided dice. 2. **Formula:** $$P(2) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}$$ 3. **Calculate:** $$P(2) = \frac{1}{6} \approx 0.167$$ 4. **Answer:** The probability of rolling a 2 rounded to the nearest thousandth is 0.167. --- 1. **State the problem:** Find the probability of picking a blue m&m from a jar with 3 red, 10 orange, 4 yellow, 6 green, 5 brown, and 4 blue m&ms. 2. **Calculate total m&ms:** $$3 + 10 + 4 + 6 + 5 + 4 = 32$$ 3. **Calculate probability:** $$P(\text{blue}) = \frac{4}{32} = \frac{1}{8} = 0.125$$ 4. **Answer:** The probability of picking a blue m&m rounded to the nearest thousandth is 0.125. --- 1. **State the problem:** Explain how to find the number of possible outcomes for a given scenario. 2. **Explanation:** To find the number of possible outcomes, list all the distinct results that can occur in the scenario. For example, when rolling a six-sided dice, the possible outcomes are {1, 2, 3, 4, 5, 6}, so there are 6 possible outcomes. If the scenario involves multiple steps or events, multiply the number of outcomes for each step to find the total number of possible outcomes (Fundamental Counting Principle). For example, flipping a coin and rolling a dice has $2 \times 6 = 12$ possible outcomes. This method helps in calculating probabilities and understanding the sample space.