Create a game (can involve a raffle, spinner, marbles in a bag, dice, etc.) that involves expected value.

Describe the game in detail.
What is the expected value of the game?  Show all your work.
Is it a fair game?  Explain why or why not.
Change one thing about your game.  Do NOT calculate the new expected value of your game

### Game: "Raffle Spinner"

#### Game Description:
- **Setup**: The game involves a spinner divided into 5 equal sections, numbered 1 through 5. Players must pay $5 to play the game, and they win a prize depending on where the spinner lands.
- **Prizes**:
  1. If the spinner lands on **1**, the player wins **$1**.
  2. If the spinner lands on **2**, the player wins **$3**.
  3. If the spinner lands on **3**, the player wins **$5**.
  4. If the spinner lands on **4**, the player wins **$10**.
  5. If the spinner lands on **5**, the player wins **$20**.

#### Goal:
- The objective is to calculate the **expected value** of the game to determine whether it is a "fair" game for the player.

---

### Expected Value of the Game

Let’s calculate the expected value (EV), which is the average amount a player can expect to win (or lose) per game in the long run.

- **Outcome 1**: Player wins **$1** with probability $ \frac{1}{5} $.
- **Outcome 2**: Player wins **$3** with probability $ \frac{1}{5} $.
- **Outcome 3**: Player wins **$5** with probability $ \frac{1}{5} $.
- **Outcome 4**: Player wins **$10** with probability $ \frac{1}{5} $.
- **Outcome 5**: Player wins **$20** with probability $ \frac{1}{5} $.

#### Calculating the Expected Value:
The formula for expected value is:

\[
\text{EV} = (P_1 \cdot W_1) + (P_2 \cdot W_2) + (P_3 \cdot W_3) + (P_4 \cdot W_4) + (P_5 \cdot W_5)
\]

Where:
- $ P_i $ is the probability of each outcome, which is $ \frac{1}{5} $ for all outcomes.
- $ W_i $ is the winning amount for each outcome.

Substitute the values:

\[
\text{EV} = \left( \frac{1}{5} \cdot 1 \right) + \left( \frac{1}{5} \cdot 3 \right) + \left( \frac{1}{5} \cdot 5 \right) + \left( \frac{1}{5} \cdot 10 \right) + \left( \frac{1}{5} \cdot 20 \right)
\]

Simplifying:

\[
\text{EV} = \frac{1}{5} \cdot (1 + 3 + 5 + 10 + 20)
\]
\[
\text{EV} = \frac{1}{5} \cdot 39 = 7.8
\]

The expected value is **$7.80**.

#### Subtract the Entry Fee:
Since the player pays $5 to play, the net expected value becomes:

\[
\text{Net EV} = 7.8 - 5 = 2.8
\]

The expected net gain for the player is **$2.80**.

---

### Is it a Fair Game?

A game is considered "fair" if the expected value is **0**. In this case, the expected value for the player is **$2.80**, meaning the player is expected to win, on average, **$2.80** per game.

- **Conclusion**: This is **not a fair game** because the player has a positive expected value. In fact, it favors the player, and the game operator (the person running the raffle) would lose money in the long run.

---

### Change One Thing About the Game

**Change**: Increase the entry fee from **$5 to $8**.

---

Would you like further details, or do you have any questions?

Here are five follow-up questions to expand on this topic:
1. What happens to the expected value if we adjust one of the prize amounts?
2. How would you calculate the variance of the game's outcomes?
3. Can you design a game with a negative expected value for the player?
4. How does the concept of expected value apply to real-world gambling games?
5. What is the long-term impact of repeatedly playing this game?

**Tip**: A game with a positive expected value can attract players, but it may not be sustainable for the game operator in the long run.

Create a game (can involve a raffle, spinner, marbles in a bag, dice, etc.) that involves expected value. Describe the game in detail. What is the expected value of the game? Show all your work. Is it a fair game? Explain why or why not. Change one thing about your game. Do NOT calculate the new expected value of your game.

Explore how to calculate the expected value and fairness of a raffle spinner game. This problem covers probability, expected value, and fairness principles, with a step-by-step solution showing the expected gain of the player. Suitable for Grades 9-12 math students.

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