1. Consider the two independent rolls of a fair 6 -sided dice. What is the probability of getting at least one 6 given the two rolls land on different numbers?

To determine the probability of getting **at least one 6** given that the **two rolls land on different numbers**, we'll use conditional probability. ### Definitions: - **Event A**: At least one of the two rolls is a 6. - **Event B**: The two rolls land on different numbers. We need to compute \( P(A \mid B) \), which is the probability of **Event A** occurring given that **Event B** has occurred. The formula for conditional probability is: \[ P(A \mid B) = \frac{P(A \cap B)}{P(B)} \] ### Step 1: Calculate \( P(B) \) - Probability that the two rolls are different. - **Total possible outcomes** when rolling two 6-sided dice: \( 6 \times 6 = 36 \). - **Favorable outcomes for different numbers**: Total outcomes minus the outcomes where both dice show the same number. There are 6 outcomes where both dice show the same number (e.g., (1,1), (2,2), ..., (6,6)). \[ P(B) = \frac{36 - 6}{36} = \frac{30}{36} = \frac{5}{6} \] ### Step 2: Calculate \( P(A \cap B) \) - Probability that at least one 6 occurs and the two rolls are different. - **Cases where at least one 6 occurs and the numbers are different**: 1. First die is 6 and the second die is not 6: \( 5 \) outcomes. 2. Second die is 6 and the first die is not 6: \( 5 \) outcomes. \[ P(A \cap B) = \frac{5 + 5}{36} = \frac{10}{36} = \frac{5}{18} \] ### Step 3: Compute the Conditional Probability \( P(A \mid B) \). \[ P(A \mid B) = \frac{P(A \cap B)}{P(B)} = \frac{\frac{5}{18}}{\frac{5}{6}} = \frac{5}{18} \times \frac{6}{5} = \frac{1}{3} \] ### Final Answer: The probability is **1⁄3**.