1. **State the problem:** We need to find the probability of rolling a double (both dice show the same number) given that the sum of the two dice is 9. 2. **Recall the formula for conditional probability:** $$P(A|B) = \frac{P(A \cap B)}{P(B)}$$ where $A$ is the event "rolling a double" and $B$ is the event "sum is 9". 3. **Identify the sample space for event $B$ (sum = 9):** Possible pairs of dice rolls that sum to 9 are: $$(3,6), (4,5), (5,4), (6,3)$$ There are 4 such outcomes. 4. **Identify the event $A \cap B$ (double and sum = 9):** A double means both dice show the same number, so possible doubles are $(1,1), (2,2), (3,3), (4,4), (5,5), (6,6)$. None of these pairs sum to 9 because: - $3+3=6$ - $4+4=8$ - $5+5=10$ So, there are no doubles that sum to 9. 5. **Calculate the probability:** $$P(A \cap B) = 0$$ $$P(B) = \frac{4}{36} = \frac{1}{9}$$ 6. **Apply the conditional probability formula:** $$P(A|B) = \frac{0}{\frac{1}{9}} = 0$$ **Final answer:** The probability of rolling a double given that the sum is 9 is $0$.