A First Course in Probability (10th Edition)
10th Edition
ISBN: 9780134753119
Author: Sheldon Ross
Publisher: PEARSON
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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](https://content.bartleby.com/qna-images/question/cd6f5f65-b970-451e-965c-fe91ac26c60e/02c2450d-888d-4483-8151-53a1873db1d7/165rxs_processed.jpeg)
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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