Let's break this problem down to understand the complement of the event in which Joe plays in the first match.
1. Identify the sample space: There are three players: Joe (J), Keitaro (K), and Luis (L). The sample space consists of all possible pairs of two players who could be drawn to play the first match. These pairs (unordered) are:
- JK (Joe and Keitaro)
- JL (Joe and Luis)
- KJ (Keitaro and Joe) - which is the same as JK
- KL (Keitaro and Luis)
- LJ (Luis and Joe) - which is the same as JL
- LK (Luis and Keitaro) - which is the same as KL
Thus, for practical purposes, the sample space is:
[tex]\[
\text{{Sample space}} = \{JK, JL, KL\}
\][/tex]
2. Determine the event where Joe plays: We list the pairs where Joe is one of the players in the first match:
- JK (Joe and Keitaro)
- JL (Joe and Luis)
Thus, the event where Joe plays in the first match is:
[tex]\[
\{JK, JL\}
\][/tex]
3. Find the complement of the event where Joe plays: The complement of this event is the set of pairs where Joe does not play. From the sample space, the remaining pairs are:
[tex]\[
\{KL\} (or equivalently, \{LK\} since order doesn't matter)
\][/tex]
4. Select the subset of the sample space representing the complement of Joe playing: The subset where Joe is not playing is:
[tex]\[
\{KL, LK\}
\][/tex]
Thus, the correct choice for the complement of the event in which Joe plays in the first match is:
[tex]\[
A = \{KL, LK\}
\][/tex]
So the correct subset is:
[tex]\[
A = \{K L, L K\}
\][/tex]
