To identify the sample space of Max's three free throw attempts, let's list all the possible outcomes of each throw. Each throw can either be a success (S) or a failure (F). Since there are three throws, we need to account for all combinations of successes and failures across these three attempts.
Here is the step-by-step process:
1. Consider the first throw:
- It can be either a success (S) or a failure (F).
2. For each outcome of the first throw, consider the second throw:
- Each outcome from the first throw still allows the second throw to be either a success (S) or a failure (F).
3. Similarly, for each combination of the first and second throws, consider the third throw:
- Each combination from the first and second throws allows the third throw to be either a success (S) or a failure (F).
We systematically list out all possible sequences of successes and failures for the three throws:
- If the first throw is a success (S), the second throw can be a success (S) or a failure (F), and the same applies for the third throw.
- If the first throw is a failure (F), the same logic applies for the second and third throws.
This results in the following complete enumeration of outcomes:
1. SSS (success, success, success)
2. SSF (success, success, failure)
3. SFS (success, failure, success)
4. SFF (success, failure, failure)
5. FSS (failure, success, success)
6. FSF (failure, success, failure)
7. FFS (failure, failure, success)
8. FFF (failure, failure, failure)
Therefore, the correct sample space for Max's three free throw attempts, covering all possible outcomes, is:
[tex]\[ \{ SSS, SSF, SFS, SFF, FSS, FSF, FFS, FFF \} \][/tex]
Among the given choices, the one that matches this sample space is:
C. [tex]\(\{SSS, SSF, SFS, SFF, FSS, FSF, FFS, FFF\}\)[/tex]
