1. **State the problem:** We have three gemstones worth 10, 100, and 1000 respectively. We draw two without replacement and want to find the expected value of the total value $H$ of the two drawn gemstones. 2. **Formula for expected value:** The expected value $E(H)$ is the sum of each possible outcome multiplied by its probability: $$E(H) = \sum (\text{value of outcome} \times \text{probability of outcome})$$ 3. **List all possible pairs and their values:** - Gemstones: 10, 100, 1000 - Possible pairs (order does not matter): - (10, 100) with total value $10 + 100 = 110$ - (10, 1000) with total value $10 + 1000 = 1010$ - (100, 1000) with total value $100 + 1000 = 1100$ 4. **Calculate probabilities of each pair:** - Total ways to choose 2 out of 3: $\binom{3}{2} = 3$ - Each pair is equally likely with probability $\frac{1}{3}$ 5. **Calculate expected value:** $$E(H) = 110 \times \frac{1}{3} + 1010 \times \frac{1}{3} + 1100 \times \frac{1}{3} = \frac{110 + 1010 + 1100}{3} = \frac{2220}{3} = 740$$ **Final answer:** The expected value of the total value $H$ of the two gemstones drawn is $740$.