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Sagot :
To determine which statements are true regarding this scenario, let's go through a detailed step-by-step solution.
### Definitions
- Sample Space: The set of all possible outcomes of a random experiment.
- Expected Value: A weighted average of all possible values that a random variable can take on, calculated using probabilities.
- Fair Game: A game is considered fair if the expected value is zero, meaning players neither gain nor lose points on average over time.
### Step-by-Step Solution
1. Sample Space:
The sample space for rolling a number cube (die) includes all the possible outcomes when you roll the cube:
[tex]$\{1, 2, 3, 4, 5, 6\}$[/tex]
Thus, the sample space is [tex]\(\{1, 2, 3, 4, 5, 6\}\)[/tex].
2. Assign Points:
According to the game rules, the points assigned are:
- Rolling an even number (2, 4, 6): [tex]\(+1\)[/tex] point.
- Rolling an odd number (1, 3, 5): [tex]\(-1\)[/tex] point.
3. Probabilities:
The probability of rolling each number on a fair six-sided die is:
[tex]$\frac{1}{6}$[/tex]
This means each outcome (1 through 6) has an equal probability of occurring.
4. Calculating Expected Value:
The expected value [tex]\(E(X)\)[/tex] is calculated as the sum of the possible outcomes multiplied by their respective probabilities.
[tex]\[ E(X) = \left(-1 \cdot \frac{1}{6}\right) + \left(1 \cdot \frac{1}{6}\right) + \left(-1 \cdot \frac{1}{6}\right) + \left(1 \cdot \frac{1}{6}\right) + \left(-1 \cdot \frac{1}{6}\right) + \left(1 \cdot \frac{1}{6}\right) \][/tex]
Let's compute this step-by-step:
- Points: [tex]\([-1, 1, -1, 1, -1, 1]\)[/tex]
- Probabilities: [tex]\(\frac{1}{6}\)[/tex] for each.
[tex]\[ E(X) = \frac{-1 + 1 - 1 + 1 - 1 + 1}{6} \][/tex]
Simplify the sum in the numerator:
[tex]\[ E(X) = \frac{(1 + 1 + 1) - (1 + 1 + 1)}{6} = \frac{0}{6} = 0 \][/tex]
Hence, the expected value for the game is [tex]\(0\)[/tex].
5. Fairness of the Game:
Since the expected value is [tex]\(0\)[/tex], this means the game is fair. A fair game implies players, on average, do not have a net gain or loss of points over time.
### True Statements
- The sample space is [tex]\(\{1,2,3,4,5,6\}\)[/tex].
This is true because these are all the possible outcomes when you roll a six-sided die.
- The expected value for the game is 0.
This is true as calculated above.
- The game is fair because the expected value is equal to 0.
This is true because a zero expected value indicates fairness.
### False Statements
- The expected value for the game is [tex]\(\frac{1}{2}\)[/tex].
This is false because the expected value is [tex]\(0\)[/tex], not [tex]\(\frac{1}{2}\)[/tex].
- The game is unfair because the expected value does not equal 0.
This is false because the expected value is indeed [tex]\(0\)[/tex].
### Conclusion
The true statements are:
- The sample space is [tex]\(\{1,2,3,4,5,6\}\)[/tex].
- The expected value for the game is [tex]\(0\)[/tex].
- The game is fair because the expected value is equal to [tex]\(0\)[/tex].
### Definitions
- Sample Space: The set of all possible outcomes of a random experiment.
- Expected Value: A weighted average of all possible values that a random variable can take on, calculated using probabilities.
- Fair Game: A game is considered fair if the expected value is zero, meaning players neither gain nor lose points on average over time.
### Step-by-Step Solution
1. Sample Space:
The sample space for rolling a number cube (die) includes all the possible outcomes when you roll the cube:
[tex]$\{1, 2, 3, 4, 5, 6\}$[/tex]
Thus, the sample space is [tex]\(\{1, 2, 3, 4, 5, 6\}\)[/tex].
2. Assign Points:
According to the game rules, the points assigned are:
- Rolling an even number (2, 4, 6): [tex]\(+1\)[/tex] point.
- Rolling an odd number (1, 3, 5): [tex]\(-1\)[/tex] point.
3. Probabilities:
The probability of rolling each number on a fair six-sided die is:
[tex]$\frac{1}{6}$[/tex]
This means each outcome (1 through 6) has an equal probability of occurring.
4. Calculating Expected Value:
The expected value [tex]\(E(X)\)[/tex] is calculated as the sum of the possible outcomes multiplied by their respective probabilities.
[tex]\[ E(X) = \left(-1 \cdot \frac{1}{6}\right) + \left(1 \cdot \frac{1}{6}\right) + \left(-1 \cdot \frac{1}{6}\right) + \left(1 \cdot \frac{1}{6}\right) + \left(-1 \cdot \frac{1}{6}\right) + \left(1 \cdot \frac{1}{6}\right) \][/tex]
Let's compute this step-by-step:
- Points: [tex]\([-1, 1, -1, 1, -1, 1]\)[/tex]
- Probabilities: [tex]\(\frac{1}{6}\)[/tex] for each.
[tex]\[ E(X) = \frac{-1 + 1 - 1 + 1 - 1 + 1}{6} \][/tex]
Simplify the sum in the numerator:
[tex]\[ E(X) = \frac{(1 + 1 + 1) - (1 + 1 + 1)}{6} = \frac{0}{6} = 0 \][/tex]
Hence, the expected value for the game is [tex]\(0\)[/tex].
5. Fairness of the Game:
Since the expected value is [tex]\(0\)[/tex], this means the game is fair. A fair game implies players, on average, do not have a net gain or loss of points over time.
### True Statements
- The sample space is [tex]\(\{1,2,3,4,5,6\}\)[/tex].
This is true because these are all the possible outcomes when you roll a six-sided die.
- The expected value for the game is 0.
This is true as calculated above.
- The game is fair because the expected value is equal to 0.
This is true because a zero expected value indicates fairness.
### False Statements
- The expected value for the game is [tex]\(\frac{1}{2}\)[/tex].
This is false because the expected value is [tex]\(0\)[/tex], not [tex]\(\frac{1}{2}\)[/tex].
- The game is unfair because the expected value does not equal 0.
This is false because the expected value is indeed [tex]\(0\)[/tex].
### Conclusion
The true statements are:
- The sample space is [tex]\(\{1,2,3,4,5,6\}\)[/tex].
- The expected value for the game is [tex]\(0\)[/tex].
- The game is fair because the expected value is equal to [tex]\(0\)[/tex].
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