1. **Problem:** Find $P(\text{even} \mid \text{at least } 12)$ when a number from 1 to 40 is chosen at random. 2. **Step 1: Define the conditional probability formula:** $$P(A \mid B) = \frac{P(A \cap B)}{P(B)}$$ where $A$ is the event "even" and $B$ is the event "at least 12". 3. **Step 2: Identify the sample space for $B$:** Numbers at least 12 are $\{12, 13, \ldots, 40\}$. Count: $40 - 12 + 1 = 29$ numbers. 4. **Step 3: Identify $A \cap B$ (even numbers at least 12):** Even numbers from 12 to 40 inclusive are $12, 14, 16, \ldots, 40$. Count: The sequence is arithmetic with first term 12, last term 40, common difference 2. Number of terms $= \frac{40 - 12}{2} + 1 = \frac{28}{2} + 1 = 14 + 1 = 15$. 5. **Step 4: Calculate the probability:** $$P(\text{even} \mid \text{at least } 12) = \frac{15}{29}$$ **Final answer:** $$\boxed{\frac{15}{29}}$$