Theoretical and Experimental Probability 1. A number cube is rolled 24 times and lands on 2 four times and on 6 three times. (3) Find the experimental probability of landing on a 2 . Find the experimental probability of not landing on a 6 . c. Compare the experimental probability you found in part a to its theoretical probability. d. Compare the experimental probability you found in part b to its theoretical probability.

To solve the problem, we will follow these steps: 1. **Extract known conditions:** - A number cube (die) is rolled 24 times. - It lands on 2 four times. - It lands on 6 three times. 2. **Calculate the experimental probabilities:** a. **Find the experimental probability of landing on a 2:** The experimental probability \( P(E) \) of an event is calculated using the formula: \[ P(E) = \frac{\text{Number of times the event occurs}}{\text{Total number of trials}} \] For landing on a 2: \[ P(2) = \frac{4}{24} \] b. **Find the experimental probability of not landing on a 6:** First, we need to find the number of times the die did not land on 6. Since it landed on 6 three times, the number of times it did not land on 6 is: \[ 24 - 3 = 21 \] Now, we can calculate the experimental probability of not landing on a 6: \[ P(\text{not } 6) = \frac{21}{24} \] 3. **Calculate the theoretical probabilities:** The theoretical probability of landing on a specific number (like 2 or 6) when rolling a fair six-sided die is: \[ P(\text{specific number}) = \frac{1}{6} \] Therefore: - Theoretical probability of landing on a 2: \[ P(2)_{theoretical} = \frac{1}{6} \] - Theoretical probability of not landing on a 6: \[ P(\text{not } 6)_{theoretical} = 1 - P(6)_{theoretical} = 1 - \frac{1}{6} = \frac{5}{6} \] 4. **Compare experimental and theoretical probabilities:** c. **Comparison for landing on a 2:** - Experimental probability \( P(2) = \frac{4}{24} = \frac{1}{6} \) - Theoretical probability \( P(2)_{theoretical} = \frac{1}{6} \) d. **Comparison for not landing on a 6:** - Experimental probability \( P(\text{not } 6) = \frac{21}{24} = \frac{7}{8} \) - Theoretical probability \( P(\text{not } 6)_{theoretical} = \frac{5}{6} \) Now, let's calculate the exact values for the experimental probabilities and provide the final comparisons. Calculating \( P(2) \) and \( P(\text{not } 6) \): 1. \( P(2) = \frac{4}{24} = \frac{1}{6} \) 2. \( P(\text{not } 6) = \frac{21}{24} = \frac{7}{8} \) Now, let's summarize the results: - **Experimental Probability of landing on a 2:** \( \frac{1}{6} \) - **Theoretical Probability of landing on a 2:** \( \frac{1}{6} \) - **Comparison:** They are equal. - **Experimental Probability of not landing on a 6:** \( \frac{7}{8} \) - **Theoretical Probability of not landing on a 6:** \( \frac{5}{6} \) - **Comparison:** The experimental probability is greater than the theoretical probability. This concludes the solution to the problem.