Patty, Quinlan, and Rashad want to be club officers. The teacher who directs the club will place their names in a hat and choose two without looking. The student whose name is chosen first will be president and the student whose name is chosen second will be vice president.
Which choice represents the sample space, [tex]$S$[/tex], for this event?
[tex]S={PQ, QP, PR, RP, QR, RQ}[/tex]
[tex]S={PQR, PRQ, QPR, QRP, RPQ, RQP}[/tex]
[tex]S={PQR}[/tex]
[tex]S={PQ, PR, QR}[/tex]
Answer (2)
List all possible pairs of students where order matters.
The possible pairs are: PQ, PR, QP, QR, RP, RQ.
The sample space S is the set of all these pairs: S = { PQ , PR , QP , QR , RP , RQ } .
The correct answer is S = { PQ , QP , PR , RP , QR , RQ } .
Explanation
Analyze the problem The problem describes a scenario where two students are chosen from a group of three (Patty, Quinlan, and Rashad) to be club officers (president and vice president). The order in which they are chosen matters because the first student selected becomes the president, and the second becomes the vice president. We need to identify the sample space, which lists all possible outcomes of this selection process.
Determine the sample space Since the order matters, we are looking for permutations of two students chosen from a group of three. We can list all possible pairs:
Patty as president, Quinlan as vice president (PQ)
Patty as president, Rashad as vice president (PR)
Quinlan as president, Patty as vice president (QP)
Quinlan as president, Rashad as vice president (QR)
Rashad as president, Patty as vice president (RP)
Rashad as president, Quinlan as vice president (RQ)
Therefore, the sample space S is {PQ, PR, QP, QR, RP, RQ}.
Identify the correct choice The sample space S for this event is the set of all possible ordered pairs of students, where the first student is the president and the second is the vice president. The correct sample space is:
S = { PQ , QP , PR , RP , QR , RQ }
Examples
In a school election, understanding the sample space helps to list all possible outcomes for president and vice president. If there are 3 candidates (A, B, C), the possible outcomes are AB, AC, BA, BC, CA, CB, where the first letter represents the president and the second represents the vice president. This ensures every possible leadership combination is considered.
The sample space for the selection of president and vice president from Patty, Quinlan, and Rashad includes all possible ordered pairs: {PQ, PR, QP, QR, RP, RQ}. Therefore, the correct option is S = {PQ, PR, QP, QR, RP, RQ}.
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