following numbers are equally distributed on a spinner 1,1,2,2,3,3,4,5,5,5. If you spin odd number you loose triple points of that number, if you spin even number you gain double points of that number. what is the expected value of this game and is this a fair game?

To solve this problem, let's break it down step by step.

### Step 1: Determine the probabilities of each outcome.
The numbers on the spinner are: 1, 1, 2, 2, 3, 3, 4, 5, 5, 5.

- There are 10 numbers in total.
- Probability of spinning a 1: \( \frac{2}{10} = 0.2 \)
- Probability of spinning a 2: \( \frac{2}{10} = 0.2 \)
- Probability of spinning a 3: \( \frac{2}{10} = 0.2 \)
- Probability of spinning a 4: \( \frac{1}{10} = 0.1 \)
- Probability of spinning a 5: \( \frac{3}{10} = 0.3 \)

### Step 2: Calculate the points for each outcome.
- **Odd Numbers (1, 3, 5)**: You lose triple the points.
  - Spinning a 1: You lose \(3 \times 1 = -3\) points.
  - Spinning a 3: You lose \(3 \times 3 = -9\) points.
  - Spinning a 5: You lose \(3 \times 5 = -15\) points.
  
- **Even Numbers (2, 4)**: You gain double the points.
  - Spinning a 2: You gain \(2 \times 2 = 4\) points.
  - Spinning a 4: You gain \(2 \times 4 = 8\) points.

### Step 3: Calculate the expected value.
The expected value \(E\) is the sum of all possible outcomes, each multiplied by its probability:

\[
E = (0.2 \times -3) + (0.2 \times 4) + (0.2 \times -9) + (0.1 \times 8) + (0.3 \times -15)
\]

Let's compute this step-by-step:

- \(0.2 \times -3 = -0.6\)
- \(0.2 \times 4 = 0.8\)
- \(0.2 \times -9 = -1.8\)
- \(0.1 \times 8 = 0.8\)
- \(0.3 \times -15 = -4.5\)

Adding these together:

\[
E = -0.6 + 0.8 - 1.8 + 0.8 - 4.5 = -5.3
\]

### Step 4: Conclusion on fairness.
The expected value of the game is \(-5.3\). Since the expected value is negative, this indicates that, on average, you will lose 5.3 points each time you play the game. Therefore, this is **not** a fair game, as a fair game would have an expected value of 0.

### Summary
- **Expected Value**: \(-5.3\)
- **Fair Game?**: No, the game is not fair.

Would you like more details or have any further questions?

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Here are 8 related questions you might find interesting:

1. How would the expected value change if the spinner had more even numbers?
2. What if you gained triple points for even numbers and lost double points for odd numbers?
3. How does the expected value change if all the numbers are equally likely?
4. Can you calculate the variance and standard deviation of the points in this game?
5. What would be the expected value if the points were gained and lost equally?
6. How does the probability distribution affect the fairness of a game?
7. Could changing the multiplier for odd or even numbers create a fair game?
8. How would introducing more numbers to the spinner affect the game's fairness?

**Tip**: When assessing fairness in games of chance, always calculate the expected value to determine whether the game is biased in favor of the player or the house.

Explore how to calculate the expected value and fairness of a probability game involving a spinner with specified rules. Understand why this game is not fair based on its negative expected value. Suitable for high school and above.

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