To solve this problem, let's follow a step-by-step approach to understand the solution:
1. Define the Sample Space: Our first step is to identify the complete list of outcomes when two names are drawn without looking:
- When Joe (J) and Keitaro (K) are drawn, the event is represented as "J K".
- When Joe (J) and Luis (L) are drawn, the event is represented as "J L".
- When Keitaro (K) and Luis (L) are drawn, the event is represented as "K L".
Therefore, the entire sample space, [tex]\( S \)[/tex], is:
[tex]\[ S = \{ J K, J L, K L \} \][/tex]
2. Identify the Event where Joe Plays: Next, we need to identify the subset of events where Joe (J) is one of the players in the first match. These include:
- Joe (J) and Keitaro (K) playing together: "J K"
- Joe (J) and Luis (L) playing together: "J L"
So, the event where Joe plays is:
[tex]\[ E = \{ J K, J L \} \][/tex]
3. Find the Complement Event: The complement of an event [tex]\( E \)[/tex] in a sample space [tex]\( S \)[/tex] includes all the outcomes in [tex]\( S \)[/tex] that are not in [tex]\( E \)[/tex]. This means we need to find all outcomes that do not include Joe (J). From our sample space [tex]\( S \)[/tex], the only outcome left is:
- Keitaro (K) and Luis (L) playing together: "K L"
Thus, the complement of the event where Joe plays, [tex]\( E' \)[/tex], is:
[tex]\[ E' = \{ K L \} \][/tex]
4. Conclusion and Correct Answer: The subset representing the complement of the event in which Joe plays in the first match is:
[tex]\[ A = \{ K L \} \][/tex]
Therefore, the correct answer is:
[tex]\[ A = \{ K L \} \][/tex]
