At a school fair, a spinner game pays 0 rupees with probability 0.5, 10 rupees with probability 0.3…

Step 1. The mean. $E(X) = 0 \times 0.5 + 10 \times 0.3 + 20 \times 0.2 = 3 + 4 = 7$ rupees.

Step 2. The mean of the squares. $E(X^2) = 0 + 100 \times 0.3 + 400 \times 0.2 = 30 + 80 = 110$.

Step 3. The variance. $\text{Var}(X) = 110 - 7^2 = 61$.

Step 4. The standard deviation. $\sqrt{61} \approx 7.81$ rupees.

Step 5. The net gain. $7 - 8 = -1$ rupee.