a) Identify the random variable X and list the elements of the random variable X in a set notation. b) Construct the probability distribution. c) Is this a fair game? Why or why not? show your work.

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### The Simple Dice Game

Consider a simple game in which you roll a single die (numbered from 1 to 6 on their faces). The game's rules are:

- If you roll a number less than 5, you **lose three times the face value** of the die.
- If you roll a 5, you gain nothing and you lose nothing.
- If you roll a 6, you **gain points equal to five times the face value** of the die.

#### Questions:

a) **Identify the random variable X and list the elements of the random variable X in a set notation.**

b) **Construct the probability distribution.**

c) **Is this a fair game? Why or why not? Show your work.**

---

### Explanation

#### a) Identifying the Random Variable X

The random variable \( X \) represents the points gained or lost based on the outcome of the die roll.

- \( X = -3 \times \text{face value} \) for rolls 1, 2, 3, and 4.
- \( X = 0 \) for roll 5.
- \( X = 5 \times \text{face value} \) for roll 6.

So, in set notation:
\[ X = \{-12, -9, -6, -3, 0, 30\} \]

#### b) Constructing the Probability Distribution

For a fair die, each face has a probability of \(\frac{1}{6}\).

- The probability of rolling each face is \(\frac{1}{6}\).

| Face Value | X (Points) | Probability |
|------------|-------------|-------------|
| 1          | -3          | \(\frac{1}{6}\) |
| 2          | -6          | \(\frac{1}{6}\) |
| 3          | -9          | \(\frac{1}{6}\) |
| 4          | -12         | \(\frac{1}{6}\) |
| 5          | 0           | \(\frac{1}{6}\) |
| 6          | 30          | \(\frac{1}{6}\) |

#### c) Determining if this is a Fair Game

To determine if the game is fair, calculate the expected value \( E(X) \) of the points.

\[ E(X
Transcribed Image Text:### The Simple Dice Game Consider a simple game in which you roll a single die (numbered from 1 to 6 on their faces). The game's rules are: - If you roll a number less than 5, you **lose three times the face value** of the die. - If you roll a 5, you gain nothing and you lose nothing. - If you roll a 6, you **gain points equal to five times the face value** of the die. #### Questions: a) **Identify the random variable X and list the elements of the random variable X in a set notation.** b) **Construct the probability distribution.** c) **Is this a fair game? Why or why not? Show your work.** --- ### Explanation #### a) Identifying the Random Variable X The random variable \( X \) represents the points gained or lost based on the outcome of the die roll. - \( X = -3 \times \text{face value} \) for rolls 1, 2, 3, and 4. - \( X = 0 \) for roll 5. - \( X = 5 \times \text{face value} \) for roll 6. So, in set notation: \[ X = \{-12, -9, -6, -3, 0, 30\} \] #### b) Constructing the Probability Distribution For a fair die, each face has a probability of \(\frac{1}{6}\). - The probability of rolling each face is \(\frac{1}{6}\). | Face Value | X (Points) | Probability | |------------|-------------|-------------| | 1 | -3 | \(\frac{1}{6}\) | | 2 | -6 | \(\frac{1}{6}\) | | 3 | -9 | \(\frac{1}{6}\) | | 4 | -12 | \(\frac{1}{6}\) | | 5 | 0 | \(\frac{1}{6}\) | | 6 | 30 | \(\frac{1}{6}\) | #### c) Determining if this is a Fair Game To determine if the game is fair, calculate the expected value \( E(X) \) of the points. \[ E(X
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