1. **Problem Statement:** We have five cards numbered 1 through 5. Three cards are randomly flipped over. We want to find the probability that the sum of the numbers on these three cards is odd. 2. **Key Concept:** The sum of numbers is odd if and only if the count of odd numbers in the sum is odd. Since the cards are numbered 1, 2, 3, 4, 5, the odd cards are 1, 3, 5 and the even cards are 2, 4. 3. **Total number of ways to choose 3 cards:** $$\binom{5}{3} = 10$$ 4. **Possible odd sum cases:** - Case 1: Choose 1 odd card and 2 even cards (since odd + even + even = odd) - Case 2: Choose 3 odd cards (since odd + odd + odd = odd) 5. **Calculate Case 1:** - Number of odd cards = 3 - Number of even cards = 2 - Choose 1 odd card: $$\binom{3}{1} = 3$$ - Choose 2 even cards: $$\binom{2}{2} = 1$$ - Total for Case 1: $$3 \times 1 = 3$$ 6. **Calculate Case 2:** - Choose 3 odd cards: $$\binom{3}{3} = 1$$ 7. **Total favorable outcomes:** $$3 + 1 = 4$$ 8. **Calculate probability:** $$\text{Probability} = \frac{\text{favorable outcomes}}{\text{total outcomes}} = \frac{4}{10} = \frac{2}{5}$$ **Final answer:** The probability that the sum of the three cards is odd is $\boxed{\frac{2}{5}}$.