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Joe, Keitaro, and Luis play tennis. To decide who will play against each other in the first match, they put their names in a hat and choose two names without looking.

What subset of the sample space, [tex]A[/tex], represents the complement of the event in which Joe plays in the first match?

A. [tex]A = \{K, L\}[/tex]

B. [tex]A = \{K, J; K, L\}[/tex]

C. [tex]A = \{K, L; L, K\}[/tex]

D. [tex]A = \{K, J; K, L; L, J\}[/tex]

Sagot :

GinnyAnsweridn
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]
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