1. **Problem:** Find the theoretical probability that a number less than 3 is rolled on a cube numbered 1 through 6. 2. **Formula:** Probability $P(E) = \frac{\text{number of favorable outcomes}}{\text{total number of possible outcomes}}$ 3. **Step 1:** Identify favorable outcomes: numbers less than 3 are 1 and 2, so 2 outcomes. 4. **Step 2:** Total possible outcomes on a cube = 6. 5. **Step 3:** Calculate probability: $$P(\text{number} < 3) = \frac{2}{6} = \frac{1}{3}$$ 6. **Step 4:** Expected number of times a number less than 3 will be rolled in $n$ trials is: $$\text{Expected} = n \times P(\text{number} < 3) = n \times \frac{1}{3}$$ 7. **Step 5:** Calculate for given trials: - For 3 trials: $$3 \times \frac{1}{3} = 1$$ - For 12 trials: $$12 \times \frac{1}{3} = 4$$ - For 30 trials: $$30 \times \frac{1}{3} = 10$$ **Final answers:** - Theoretical probability of rolling a number less than 3 is $\frac{1}{3}$. - Expected rolls less than 3 in 3 trials: 1 - Expected rolls less than 3 in 12 trials: 4 - Expected rolls less than 3 in 30 trials: 10 *Note: Experimental probability and comparisons require actual roll data, which is not provided here.*