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, A, represents the complement of the event in which Joe plays in the first match?
A. $A={KL}$
B. $A={KJ, KL}$
C. $A={KL, LK}$
D. $A={KJ, KL, LJ}$
Answer (2)
List all possible pairs: JK, JL, KL.
Identify the pairs where Joe does not play: KL.
The complement of the event where Joe plays is the set containing only the pair KL.
The subset A representing this complement is: A = { K L }
Explanation
Analyze the problem Let's break down the problem. We have three players: Joe, Keitaro, and Luis. We want to find the set of outcomes where Joe does not play in the first match. First, we need to list all possible pairs of players.
List all possible pairs The possible pairs are: Joe and Keitaro (JK), Joe and Luis (JL), and Keitaro and Luis (KL).
Identify pairs without Joe We are looking for the complement of the event where Joe plays. This means we want the pairs where Joe is not included. From our list, only the pair Keitaro and Luis (KL) does not include Joe.
Determine the complement Therefore, the subset of the sample space, A, that represents the complement of the event in which Joe plays in the first match is A = {KL}.
Examples
This type of problem is useful in scenarios where you need to determine the probability of certain events not occurring. For example, if you're organizing a tournament and want to know the likelihood that a specific player won't be in a particular match, you would use similar logic to determine the possible combinations and their probabilities.
The complement of the event where Joe plays in the first match consists of the situation where he does not play at all. This only occurs in the match between Keitaro and Luis. Therefore, the correct answer is A = {KL}.
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