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.

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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.

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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.

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Change One Thing About the Game

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

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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.