To solve this problem, we'll determine the sample mean of the outcomes and then identify the probability associated with this mean.
### Step-by-Step Solution:
1. Identify the Outcomes:
We're given the outcomes [tex]\(2, 0, 1\)[/tex].
2. Calculate the Sample Mean:
The sample mean [tex]\(\bar{x}\)[/tex] is calculated by summing the outcomes and then dividing by the number of outcomes.
[tex]$ \bar{x} = \frac{\sum \text{outcomes}}{\text{number of outcomes}} $[/tex]
For our specific outcomes:
[tex]$ \bar{x} = \frac{2 + 0 + 1}{3} $[/tex]
[tex]$ \bar{x} = \frac{3}{3} $[/tex]
[tex]$ \bar{x} = 1 $[/tex]
3. Given Probabilities:
We need to check the probabilities provided in the model:
- [tex]\( p = 0.5 \)[/tex]
- [tex]\( p = 0.02 \)[/tex]
- [tex]\( p = 0.4 \)[/tex]
4. Determine the Correct Probability:
We need to find which probability matches the outcome and the sample mean. According to the problem statement, each probability can be considered for the exact sample mean.
5. Conclusion:
The correct tuple where the sample mean [tex]\(\bar{x}\)[/tex] is [tex]\(1\)[/tex] and the probability [tex]\(p\)[/tex] is [tex]\(0.5\)[/tex].
The correct answer to the problem is:
[tex]\( (\bar{x} = 1, p = 0.5) \)[/tex].
Therefore, the outcome 2, 0, 1 has a sample mean of [tex]\(1\)[/tex] and a probability of [tex]\(0.5\)[/tex].
