1. **State the problem:** We have a probability distribution with amounts won and their probabilities: - Amounts: 0, -6a, 2 - Probabilities: 33%, 33%, 33% We want to understand the expected value (mean) and possibly use the Z-score formula given as $$Z = \frac{x - \mu}{\sigma}$$ where $\mu$ is the mean and $\sigma$ is the standard deviation. 2. **Calculate the expected value $\mu$:** The expected value is the sum of each amount multiplied by its probability. $$\mu = 0 \times \frac{33}{100} + (-6a) \times \frac{33}{100} + 2 \times \frac{33}{100}$$ Simplify: $$\mu = 0 - \frac{198a}{100} + \frac{66}{100} = -1.98a + 0.66$$ 3. **Calculate the fraction multiplication:** Given $$\frac{4}{12} \times \frac{1}{7}$$ Simplify numerator and denominator: $$\frac{\cancel{4}}{\cancel{12}} \times \frac{1}{7} = \frac{1}{3} \times \frac{1}{7} = \frac{1}{21}$$ 4. **Use the Z-score formula:** Given $n=12$, and the formula $$Z = \frac{x - \mu}{\sigma}$$, you can calculate the Z-score for a value $x$ once $\mu$ and $\sigma$ are known. Since $\sigma$ (standard deviation) is not provided, it cannot be calculated here. **Final answers:** - Expected value: $$\mu = -1.98a + 0.66$$ - Fraction multiplication: $$\frac{4}{12} \times \frac{1}{7} = \frac{1}{21}$$