To determine the subset of the sample space that represents the complement of the event in which Joe plays in the first match, we first need to clarify the events:
- The sample space for the first match would consist of all possible pairs of players that can be chosen from Joe (J), Keitaro (K), and Luis (L). The possible pairs are:
- JK (Joe and Keitaro)
- JL (Joe and Luis)
- KL (Keitaro and Luis)
Let's denote these pairs in the sample space S:
- S = {JK, JL, KL}
- The event in which Joe plays in the first match includes the pairs where Joe is one of the players:
- Event where Joe plays: {JK, JL}
- The complement of this event, denoted as \( A' \), is the event where Joe does not play in the first match. Therefore, it consists of the outcomes where neither JK nor JL is chosen, which would leave us with:
- A' = {KL}
So, now let's look through the answer choices given to find the subset of the sample space that represents the complement of the event in which Joe plays in the first match.
- A = {KL} → This is correct as it consists of the pair where Joe does not play.
- A = {KJ, KL} → Incorrect, as it includes a pair (KJ) where Joe plays.
- A = {KL, LK} → Incorrect as it includes pair KL (which is okay) but LK is not a valid pair with the players Joe, Keitaro, and Luis.
- A = {KJ, KL, LJ} → Incorrect, as it also contains pairs where Joe plays (KJ, LJ).
Thus, the correct answer is: A = {KL}