To calculate the expected value for this scenario, the formula for the expected value $E(X)$ is used:

$E(X) = \sum_{i=1}^n x_i \cdot P(x_i)$

Where $x_i$ is the outcome value and $P(x_i)$ is the probability of that outcome. From the table provided:

Step-by-step Calculation:

1. Multiply each $x_i$ by its corresponding probability $P(x_i)$:

   • For $x_i = 0.25$ and $P(x_i) = 0.4$: $0.25 \cdot 0.4 = 0.10$
   • For $x_i = 0.25$ and $P(x_i) = 0.3$: $0.25 \cdot 0.3 = 0.075$
   • For $x_i = 0$ and $P(x_i) = 0.2$: $0 \cdot 0.2 = 0$
   • For $x_i = -0.25$ and $P(x_i) = 0.1$: $-0.25 \cdot 0.1 = -0.025$

2. Add these products together to find the expected value: $E(X) = 0.10 + 0.075 + 0 + (-0.025) = 0.15$

Final Answer:

The expected value of the scenario is $0.15.

Would you like a deeper explanation or have additional questions? Here are 5 related questions for exploration:

1. What does the expected value represent in probability and statistics?
2. How would the expected value change if one of the probabilities increased or decreased?
3. How is the concept of expected value applied in real-world decision-making?
4. Can you calculate the variance and standard deviation for this scenario?
5. How do probabilities affect the distribution of outcomes in this problem?

Tip: Always double-check that the sum of probabilities equals 1 when calculating the expected value.