To determine the sample space for rolling a fair seven-sided die, let's consider the following:
- A seven-sided die has faces numbered from 1 to 7.
- The sample space [tex]\( S \)[/tex] is the set of all possible outcomes when the die is rolled.
Given this, we can list out all the individual outcomes:
There are 7 possible outcomes:
1, 2, 3, 4, 5, 6, 7
So, the sample space [tex]\( S \)[/tex] is the set of these outcomes:
[tex]\[ S = \{1, 2, 3, 4, 5, 6, 7\} \][/tex]
This is the complete set of all possible results that can occur when rolling a fair seven-sided die. Therefore, the correct sample space is:
[tex]\[ S = \{1, 2, 3, 4, 5, 6, 7\} \][/tex]
Other provided options include:
- [tex]\( S = \{1, 2, 3, 4, 5, 6, 7, 8\} \)[/tex], which incorrectly includes the value 8, not present in a seven-sided die.
- [tex]\( S = \{1\} \)[/tex], which only lists one of the possible outcomes.
- [tex]\( S = \{7\} \)[/tex], which also only lists one of the possible outcomes.
Thus, the correct sample space for rolling a fair seven-sided die is:
[tex]\[ S = \{1, 2, 3, 4, 5, 6, 7\} \][/tex]
