To determine the sample space of this probability experiment, we need to consider all possible outcomes or combinations of choices between roommate A and roommate B.
In rock-paper-scissors, there are three possible choices: rock, paper, or scissors. Since each roommate has three choices, the total number of outcomes or sample points is the product of the number of choices for each roommate.
In this case, roommate A can choose rock, paper, or scissors, and roommate B can also choose rock, paper, or scissors. Thus, the sample space consists of all possible combinations of their choices.
To find the number of outcomes in the sample space, we multiply the number of choices for each roommate. Therefore, the correct answer is 3 choices for roommate A multiplied by 3 choices for roommate B, which gives us a total of 9 outcomes in the sample space.
Now, let's consider the event A, which represents the event that roommate A wins the game and gets to sit in the front seat. We need to determine which sample points are included in this event.
To determine the sample points in event A, we need to look for the outcomes where roommate A wins. In rock-paper-scissors, rock beats scissors, scissors beat paper, and paper beats rock.
From the given options:
a. A = {(Ra, Sb), (Pa, Rb), (Sa, Pb)}
b. A = {(Pa, Rb), (Sa, Pb)}
c. A = {(Ra, Rb), (Ra, Sb), (Pa, Rb), Sa, Pb)}
d. A = {(Ra, Pb), (Pa, Sb), (Sa, Rb)}
In event A, roommate A wins if they choose paper (Pa) and roommate B chooses rock (Rb), or if roommate A chooses scissors (Sa) and roommate B chooses paper (Pb). Thus, the correct answer is option b: A = {(Pa, Rb), (Sa, Pb)}. These two sample points represent the outcomes where roommate A wins the game.
Therefore, event A is composed of {(Pa, Rb), (Sa, Pb)}.