The task requires calculating the expected value of a lottery game described in the problem and comparing it to a fixed amount of €129 to determine the optimal choice for maximizing expected winnings.

Step 1: Formula for Expected Value

The expected value $E(G)$ of a random variable $G$ is calculated as: $E(G) = \sum_{i} g_i \cdot P(G = g_i)$ where $g_i$ represents the possible winnings, and $P(G = g_i)$ is the corresponding probability.

Step 2: Values from the Table

From the table: $g = \{40, 90, 100, 180, 300\}$ $P(G = g) = \{0.28, 0.23, 0.24, 0.20, 0.05\}$

Step 3: Calculation

We calculate $E(G)$ as follows: $E(G) = (40 \cdot 0.28) + (90 \cdot 0.23) + (100 \cdot 0.24) + (180 \cdot 0.20) + (300 \cdot 0.05)$

Let’s compute this step-by-step.

1. $40 \cdot 0.28 = 11.2$
2. $90 \cdot 0.23 = 20.7$
3. $100 \cdot 0.24 = 24.0$
4. $180 \cdot 0.20 = 36.0$
5. $300 \cdot 0.05 = 15.0$

Adding these: $E(G) = 11.2 + 20.7 + 24.0 + 36.0 + 15.0 = 106.9$

Step 4: Comparison

• Fixed winnings: €129
• Expected value of the lottery: €106.9

Since €129 > €106.9, the participant should choose the fixed winnings of €129 to maximize their expected gain.

Would you like me to explain further or provide details about the steps?

Related Questions

1. What is the significance of the expected value in decision-making under uncertainty?
2. How would the decision change if the fixed amount were less than the expected value?
3. Can variance or risk considerations affect this decision? How?
4. How does the probability distribution shape impact the expected value?
5. What happens if one of the probabilities changes? How would it alter the outcome?

Tip

When dealing with probabilities and expected values, ensure all probabilities sum to 1 as a consistency check.