Question

1. a car dealership has various combinations of vehicles in their car lot, all of which have an equally likely chance of being selected. mark each statement as true or false. correct any false statements.

______ a. there are 9 total possible outcomes.
______ b. the probability of choosing a truck that is silver or black with a leather interior is \\(\frac{1}{12}\\).
______ c. the probability of choosing a sedan with cloth interior that is not red is \\(\frac{1}{24}\\).
\

$$\begin{tabular}{|c|c|c|} \\hline vehicle & exterior & interior \\\\ \\hline sedan & red & leather \\\\ suv & white & cloth \\\\ truck & silver & \\\\ & black & \\\\ \\hline \\end{tabular}$$

Explanation:

Step 1: Analyze part a

First, we determine the number of vehicle types, exterior colors, and interior options. Vehicle types: 3 (Sedan, SUV, Truck). Exterior colors: 4 (Red, White, Silver, Black). Interior options: 2 (Leather, Cloth) for most, but Truck's interior is missing? Wait, Sedan: Red - Leather; SUV: White - Cloth; Truck: Silver, Black -? Wait, maybe the table is structured as Vehicle (3 types), Exterior (4 colors), Interior (2 types, but maybe some combinations). Wait, actually, to find total outcomes, we multiply the number of vehicle types, exterior colors, and interior options. Wait, Vehicle: 3 (Sedan, SUV, Truck). Exterior: 4 (Red, White, Silver, Black). Interior: Let's see, Sedan has Leather, SUV has Cloth, Truck has no interior listed? Wait, maybe the table is:

- Sedan: Exterior Red, Interior Leather
- SUV: Exterior White, Interior Cloth
- Truck: Exterior Silver, Interior?; Exterior Black, Interior?

Wait, maybe the problem is that each vehicle can have each exterior and interior? No, the table is probably a list where each vehicle (Sedan, SUV, Truck) can have each exterior color, and each interior. Wait, no, the original table:

VEHICLE | EXTERIOR | INTERIOR
Sedan | Red | Leather
SUV | White | Cloth
Truck | Silver |
Truck | Black |

Wait, maybe that's a typo, and Truck has two exteriors (Silver, Black) and maybe interior? Wait, maybe the correct way is: Vehicle types: 3 (Sedan, SUV, Truck). Exterior colors: 4 (Red, White, Silver, Black). Interior: 2 (Leather, Cloth). But Sedan is Red - Leather, SUV is White - Cloth, Truck has Silver and Black, but what interior? Maybe the problem is that each vehicle (3) can have each exterior (4) and each interior (2), but some are specified. Wait, no, the first step is to find total possible outcomes. Let's re-express:

Vehicle: 3 options (Sedan, SUV, Truck)

Exterior: 4 options (Red, White, Silver, Black)

Interior: 2 options (Leather, Cloth)

But wait, Sedan is Red - Leather, SUV is White - Cloth, Truck has Silver and Black, but maybe the interior for Truck is missing? Wait, maybe the problem is that the total outcomes are 3 (vehicles) * 4 (exteriors) * 2 (interiors) = 24? Wait, no, the first statement says "There are 9 total possible outcomes." Let's check: 3 vehicles, 3 exteriors? Wait, the exterior column has Red, White, Silver, Black – 4 colors. Wait, maybe the table is misformatted. Alternatively, maybe the vehicle types are 3, exterior colors 3 (Red, White, Silver) and Black is a typo? No, the user's table shows Truck with Silver and Black. Wait, maybe the correct total outcomes: Let's list all possible combinations.

Sedan:
- Red - Leather
- Red - Cloth (but table says Red - Leather)
Wait, no, maybe the table is a list of existing combinations, but the problem says "various combinations" with equally likely chance. Wait, maybe the intended is: Vehicle (3) * Exterior (4) * Interior (2) = 3*4*2=24? Wait, but statement a says 9. So 3 vehicles, 3 exteriors, 1 interior? No. Wait, maybe the table is:

Vehicle: 3 (Sedan, SUV, Truck)

Exterior: 3 (Red, White, Silver) and Black is extra? No, the table has 4 exteriors. Wait, maybe the problem is that the total outcomes are 3 (vehicles) * 3 (exteriors: Red, White, Silver) * 1 (interior? No). This is confusing. Wait, maybe the first step is to calculate total outcomes. Let's assume that Vehicle: 3, Exterior: 4, Interior: 2, so 3*4*2=24. But statement a says 9, which would be 3*3*1, but that's not. Wait, maybe the table is:

Sedan: Red - Leather

SUV: White - Cloth

Truck: Silver -?, Black -?

Wait, maybe the problem is that each vehicle (3) has 3 exteriors (Red, White, Silver) and 1 interior, but Black is extra. No, this is unclear. Wait, maybe the correct approach is:

For part a: Total possible outcomes. Let's count the number of rows in the table. The table has:

1. Sedan - Red - Leather

2. SUV - White - Cloth

3. Truck - Silver -?

4. Truck - Black -?

Wait, maybe the interior for Truck is missing, but maybe the problem is that each vehicle (3) can have each exterior (4) and each interior (2), but some are given. Wait, no, the first step is to find total outcomes. Let's suppose that Vehicle: 3, Exterior: 4, Interior: 2, so 3*4*2=24. But statement a says 9, which is 3*3*1. So maybe the table is:

Vehicle: 3 (Sedan, SUV, Truck)

Exterior: 3 (Red, White, Silver)

Interior: 1 (Leather for Sedan, Cloth for SUV,? for Truck). No, this is not working. Wait, maybe the original problem has a table where Vehicle: 3, Exterior: 3, Interior: 1, but Black is an extra. I think there's a typo, but let's proceed.

Wait, maybe the correct total outcomes: 3 vehicles, 3 exteriors (Red, White, Silver), and 1 interior? No. Alternatively, the table is:

Sedan: Red - Leather

SUV: White - Cloth

Truck: Silver - Leather

Truck: Black - Cloth

Then total outcomes: 4? No. This is confusing. Wait, maybe the problem is that the total number of possible outcomes is calculated by multiplying the number of vehicle types, exterior colors, and interior options. Let's assume Vehicle: 3, Exterior: 4, Interior: 2, so 3*4*2=24. But statement a says 9, which is false. Wait, maybe the table is:

Vehicle: 3 (Sedan, SUV, Truck)

Exterior: 3 (Red, White, Silver)

Interior: 1 (Leather for Sedan, Cloth for SUV, Leather for Truck). Then total outcomes: 3*3*1=9. Ah, that makes sense. So Sedan: Red - Leather, White - Leather, Silver - Leather; SUV: Red - Cloth, White - Cloth, Silver - Cloth; Truck: Red - Leather, White - Leather, Silver - Leather? No, that's not matching the table. Wait, the table has Sedan - Red - Leather, SUV - White - Cloth, Truck - Silver, Truck - Black. Maybe the exterior colors are Red, White, Silver, Black (4), vehicle types 3, interior 2, so 3*4*2=24. So statement a says 9, which is false. But maybe I'm overcomplicating. Let's move to part b.

Part b: Probability of choosing a truck that is silver or black with a leather interior. First, find the number of favorable outcomes. Truck, exterior silver or black, interior leather. How many such outcomes? Truck has two exteriors (silver, black), interior leather: 2 outcomes. Total outcomes: let's say 24 (3*4*2). So probability is 2/24 = 1/12. So statement b: is it true? If total outcomes are 24, then 2/24=1/12, so true? Wait, but maybe total outcomes are 9. No, this is confusing.

Part c: Probability of choosing a sedan with cloth interior that is not red. Sedan, cloth interior, not red (exterior white, silver, black). Number of favorable outcomes: Sedan, exterior (white, silver, black) – 3 options, interior cloth – 1 option. So 3*1=3. Total outcomes: 24, so probability 3/24=1/8. But statement c says 1/24, which is false. Wait, but maybe total outcomes are 24.

Wait, maybe the correct approach is:

First, determine the number of vehicle types (V=3), exterior colors (E=4), interior options (I=2). So total outcomes = V*E*I = 3*4*2=24.

Part a: Total outcomes 9? 3*3*1=9. So 24≠9, so statement a is false. Correct: 24 total outcomes.

Part b: Truck, exterior silver or black (2 colors), interior leather (1 option). So number of favorable: 1 (truck) * 2 (silver, black) * 1 (leather) = 2. Probability: 2/24=1/12. So statement b is true.

Part c: Sedan, cloth interior, not red (exterior white, silver, black – 3 colors). Number of favorable: 1 (sedan) * 3 (exteriors) * 1 (cloth) = 3. Probability: 3/24=1/8. Statement c says 1/24, so false. Correct: 1/8.

But let's re-express:

Step 1: Part a

To find total possible outcomes, we consider the number of vehicle types (3: Sedan, SUV, Truck), exterior colors (4: Red, White, Silver, Black), and interior options (2: Leather, Cloth). Using the multiplication principle, total outcomes = 3 (vehicles) * 4 (exteriors) * 2 (interiors) = 24. The statement claims 9, so it is false. Correct: There are 24 total possible outcomes.

Step 2: Part b

Favorable outcomes: Truck (1 vehicle) * 2 exteriors (Silver, Black) * 1 interior (Leather) = 2. Total outcomes = 24. Probability = 2/24 = 1/12. So the statement is true.

Step 3: Part c

Favorable outcomes: Sedan (1 vehicle) * 3 exteriors (White, Silver, Black – not red) * 1 interior (Cloth) = 3. Total outcomes = 24. Probability = 3/24 = 1/8. The statement claims 1/24, so it is false. Correct: The probability is 1/8.

Answer:

a. False (Correct: 24 total outcomes)
b. True
c. False (Correct: Probability is $\frac{1}{8}$)