1. **Problem Statement:** Calculate the mean, variance, and standard deviation of the given probability distribution for the prizes. 2. **Formulas and Important Rules:** - The mean (expected value) $\mu$ of a discrete random variable $X$ is given by: $$\mu = E(X) = \sum x_i p_i$$ where $x_i$ are the values and $p_i$ their probabilities. - The variance $\sigma^2$ is: $$\sigma^2 = Var(X) = \sum (x_i - \mu)^2 p_i$$ - The standard deviation $\sigma$ is the square root of the variance: $$\sigma = \sqrt{Var(X)}$$ 3. **Calculate the Mean:** $$\mu = (0)(0.40) + (10)(0.35) + (20)(0.20) + (50)(0.05)$$ $$\mu = 0 + 3.5 + 4 + 2.5 = 10$$ 4. **Calculate the Variance:** First, compute each squared deviation times probability: $$ (0 - 10)^2 (0.40) = 100 \times 0.40 = 40$$ $$ (10 - 10)^2 (0.35) = 0 \times 0.35 = 0$$ $$ (20 - 10)^2 (0.20) = 100 \times 0.20 = 20$$ $$ (50 - 10)^2 (0.05) = 1600 \times 0.05 = 80$$ Sum these: $$\sigma^2 = 40 + 0 + 20 + 80 = 140$$ 5. **Calculate the Standard Deviation:** $$\sigma = \sqrt{140} \approx 11.83$$ **Final answers:** - Mean prize value $\mu = 10$ - Variance $\sigma^2 = 140$ - Standard deviation $\sigma \approx 11.83$