To solve this problem, we need to calculate the expected value of Milan's game based on the outcomes and probabilities.

Step-by-Step Solution:

(a) Find the expected value of playing the game

1. Outcomes and payouts:

   • $1 for number 1.
   • $3 for number 2.
   • $5 for number 3.
   • $7 for number 4.
   • -$8 (loss) for numbers 5 and 6.

2. Probabilities:

   • Since the spinner is fair with 6 equal-sized slices, each outcome has a probability of $\frac{1}{6}$.

3. Expected value formula: The expected value is calculated as: $E(X) = \sum (\text{payout} \times \text{probability})$

   Substitute the values: $E(X) = \left(1 \cdot \frac{1}{6}\right) + \left(3 \cdot \frac{1}{6}\right) + \left(5 \cdot \frac{1}{6}\right) + \left(7 \cdot \frac{1}{6}\right) + \left(-8 \cdot \frac{1}{6}\right) + \left(-8 \cdot \frac{1}{6}\right)$

4. Simplify: $E(X) = \frac{1}{6}(1 + 3 + 5 + 7 - 8 - 8)$

   $E(X) = \frac{1}{6}(0)$

   $E(X) = 0$

The expected value of playing the game is $0.

(b) What can Milan expect in the long run?

Since the expected value is $0$, this means that Milan can expect to break even in the long run, neither gaining nor losing money on average.

Final Answers:

1. (a) Expected value = $\mathbf{0}$ dollars.
2. (b) Milan can expect to break even (neither gain nor lose money).

Would you like to explore any details or additional explanations?
Here are 5 related questions for practice:

1. What is the interpretation of the expected value in probability?
2. How does changing one payout affect the expected value?
3. If the spinner is biased, how would the expected value change?
4. What is the variance and standard deviation of this game?
5. How can we verify if the game is fair using simulations?

Tip: When calculating expected value, always multiply each outcome by its probability before summing!