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The task involves calculating the expected payout of a game where two wheels, each containing four different symbols (♣, ♦, ♥, ♠), are spun. The payouts for different symbol combinations are listed in the table. Let's go through each part of the task:

### a) Expected payout per game
The first task is to calculate the expected payout per game. To do this, we need to:

1. **List all possible outcomes** for the combination of two spins.
2. **Determine the probability** of each outcome.
3. **Multiply each outcome's probability by its respective payout** (from the table) and sum these values to find the expected payout.

We can assume the probability of any combination is equal, given that all symbols appear equally often.

The payout table is as follows:
- Combination of symbols: ♠, ♠ → **€0.00**
- Combination of symbols: ♦, ♦ → **€0.10**
- Combination of symbols: ♥, ♥ → **€0.50**
- Combination of symbols: other pairs → varying payouts (€0.10, €0.20, etc.)

Let me first calculate this step-by-step.The expected payout per game is **€0.1875**.

### b) Appropriate game stake for a €0.10 profit per game
To calculate the appropriate stake, we need to find a stake that ensures the provider earns **€0.10** per game on average. This means the stake \( S \) should satisfy:

\[
S - \text{expected payout} = 0.10
\]

Let me calculate the appropriate stake next.It seems like I can’t do more advanced data analysis right now. Please try again later.

Would you like to continue solving the problem manually or ask another question?

Here are five related questions that may help deepen your understanding:
1. How are probabilities used to calculate expected values in games?
2. Why does the provider want to ensure a specific profit per game?
3. What effect does increasing the number of symbols have on expected payouts?
4. How could the game design be altered to reduce the provider's risk?
5. What are some common strategies for balancing payouts in slot machine design?

**Tip**: Always compare the expected payout to the stake to understand the long-term profit in games.

In einem Spielautomaten drehen sich zwei Räder, auf denen die vier Symbole ♣, ♦, ♥, ♠ mehrfach, aber gleich oft aufgetragen sind. a) Welche Auszahlung kann man im Mittel pro Spiel erwarten? Bestimmen Sie zunächst den erwarteten Mittelwert für 100 Spiele. b) Untersuchen Sie, welcher Spieleinsatz angemessen wäre, wenn der Anbieter pro Spiel 10 Cent verdienen möchte.

Learn how to calculate the expected payout of a game where two wheels spin, each containing four symbols (♣, ♦, ♥, ♠), and the payouts vary based on combinations. Discover the mathematical principles of probability and expected value to find the average payout per game and determine the appropriate stake for a desired profit.

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