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Let's solve the problem step-by-step:

### Given Information
1. Michael rolls a fair six-sided number cube (dice) with numbers 1 through 6.
2. Payouts based on the number rolled:
   - Wins $1 if a 1 is rolled.
   - Wins $2 if a 2 is rolled.
   - Wins $3 if a 3 is rolled.
   - Wins $4 if a 4 is rolled.
   - **Loses $2.75** if a 5 or 6 is rolled.

### Part (a): Expected Value of Playing the Game
To calculate the expected value $E$, we multiply each outcome by its probability and sum them up:

1. Probability of each number (1 through 6) is $\frac{1}{6}$.
2. Expected value calculation:
   \[
   E = \left(\frac{1}{6} \times 1\right) + \left(\frac{1}{6} \times 2\right) + \left(\frac{1}{6} \times 3\right) + \left(\frac{1}{6} \times 4\right) + \left(\frac{1}{6} \times (-2.75)\right) + \left(\frac{1}{6} \times (-2.75)\right)
   \]
3. Breaking it down:
   - For rolls 1 through 4: $\frac{1}{6} \times (1 + 2 + 3 + 4) = \frac{1}{6} \times 10 = 1.6667$.
   - For rolls 5 and 6: $\frac{1}{6} \times (-2.75) + \frac{1}{6} \times (-2.75) = \frac{1}{6} \times (-5.5) = -0.9167$.
4. Adding these together:
   \[
   E = 1.6667 - 0.9167 = 0.75
   \]

**Answer to (a):** The expected value of playing the game is **0.75 dollars**.

### Part (b): Long-Run Expectation
Since the expected value is positive (0.75 dollars), Michael can expect to **gain money** in the long run.

**Answer to (b):**
- **Michael can expect to gain money.**
- **He can expect to win 0.75 dollars per roll.**

Would you like further details on this calculation or have any questions?

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### Related Questions

1. How would the expected value change if the loss for rolling a 5 or 6 was increased to $3?
2. What would the expected value be if Michael only lost $1 for rolling a 5 or 6?
3. How would the outcome change if Michael rolled a weighted dice where the probability of rolling a 6 is doubled?
4. What is the variance of Michael’s winnings in this game?
5. How would the expected value change if Michael only wins if he rolls an odd number?

### Tip
In probability calculations involving games, expected value gives a fair estimate of long-term outcomes, but variance is crucial to understand short-term risks.

Michael is playing a game of chance in which he rolls a number cube with sides numbered from 1 to 6. The number cube is fair, so a side is rolled at random. He wins $1 if a 1 is rolled, $2 if a 2 is rolled, $3 if a 3 is rolled, and $4 if a 4 is rolled. He loses $2.75 if a 5 or 6 is rolled. (a) Find the expected value of playing the game. (b) What can Michael expect in the long run, after playing the game many times?

Learn how to calculate the expected value in a game of chance where Michael rolls a fair six-sided dice with varying payouts. This problem explores the use of expected value to predict long-term gains or losses, making it suitable for high school students.

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