To determine the sample space for rolling a fair eight-sided die, we need to understand what the sample space represents. The sample space is the set of all possible outcomes of an experiment.
In the case of rolling a fair eight-sided die, the die has 8 faces, each showing a unique number from 1 to 8. Each of these faces represents a possible outcome when the die is rolled. Therefore, the sample space must include every possible result that can appear on the die.
Let's examine the options provided:
1. [tex]\( S = \{1\} \)[/tex]: This set only includes the number 1, which does not cover all possible outcomes of rolling the die.
2. [tex]\( S = \{8\} \)[/tex]: This set only includes the number 8, similarly failing to cover all possible outcomes.
3. [tex]\( S = \{1, 2, 3, 4, 5, 6\} \)[/tex]: This set covers numbers 1 through 6 but misses the numbers 7 and 8.
4. [tex]\( S = \{1, 2, 3, 4, 5, 6, 7, 8\} \)[/tex]: This set includes all numbers from 1 through 8, encompassing every possible outcome of rolling the die.
Thus, the correct sample space for rolling a fair eight-sided die is:
[tex]\[ S = \{1, 2, 3, 4, 5, 6, 7, 8\} \][/tex]
So, the sample space is:
[tex]\[ \boxed{\{1, 2, 3, 4, 5, 6, 7, 8\}} \][/tex]
