1. **Problem 1:** Two balls are drawn in succession without replacement from an urn containing 5 orange balls and 6 violet balls. Let $V$ be the random variable representing the number of violet balls drawn. Find the values of $V$ and construct its probability distribution. 2. **Step 1: Define the possible values of $V$.** Since two balls are drawn, $V$ can be 0, 1, or 2. 3. **Step 2: Calculate total number of ways to draw 2 balls from 11 balls.** Total balls = 5 orange + 6 violet = 11 Number of ways to choose 2 balls: $$\binom{11}{2} = \frac{11 \times 10}{2} = 55$$ 4. **Step 3: Calculate probabilities for each value of $V$.** - $V=0$ (no violet balls): both balls are orange. Number of ways: $$\binom{5}{2} = \frac{5 \times 4}{2} = 10$$ Probability: $$P(V=0) = \frac{10}{55} = \frac{2}{11}$$ - $V=1$ (one violet ball): one violet and one orange ball. Number of ways: $$\binom{6}{1} \times \binom{5}{1} = 6 \times 5 = 30$$ Probability: $$P(V=1) = \frac{30}{55} = \frac{6}{11}$$ - $V=2$ (two violet balls): both balls are violet. Number of ways: $$\binom{6}{2} = \frac{6 \times 5}{2} = 15$$ Probability: $$P(V=2) = \frac{15}{55} = \frac{3}{11}$$ 5. **Step 4: Verify probabilities sum to 1.** $$\frac{2}{11} + \frac{6}{11} + \frac{3}{11} = \frac{11}{11} = 1$$ 6. **Step 5: Construct the probability distribution table.** | $V$ | 0 | 1 | 2 | |-----|---|---|---| | $P(V)$ | $\frac{2}{11}$ | $\frac{6}{11}$ | $\frac{3}{11}$ | --- **Final answer:** The random variable $V$ takes values 0, 1, and 2 with probabilities $\frac{2}{11}$, $\frac{6}{11}$, and $\frac{3}{11}$ respectively. --- **Note:** The second problem is not solved as per instructions to solve only the first problem.