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To determine the probability that a randomly chosen customer has purchased either a truck or a white vehicle, we will follow these detailed steps:
Step 1: Calculate the total number of purchases.
We will sum all the purchases for each type of car and color:
[tex]\[ \begin{aligned} &\text{Total purchases} = \\ &\text{Red Sedan} + \text{Red SUV} + \text{Red Truck} + \\ &\text{Blue Sedan} + \text{Blue SUV} + \text{Blue Truck} + \\ &\text{White Sedan} + \text{White SUV} + \text{White Truck} \\ &= 17 + 7 + 3 \\ &+ 17 + 19 + 23 \\ &+ 43 + 37 + 53 \\ &= 219 \\ \end{aligned} \][/tex]
Step 2: Calculate the total number of trucks.
Sum all the truck purchases for each color:
[tex]\[ \begin{aligned} &\text{Total Trucks} =\\ &\text{Red Truck} + \text{Blue Truck} + \text{White Truck}\\ &= 3 + 23 + 53 \\ &= 79 \end{aligned} \][/tex]
Step 3: Calculate the total number of white vehicles.
Sum all the purchases of white vehicles (Sedan, SUV, and Truck):
[tex]\[ \begin{aligned} &\text{Total White Vehicles} = \\ &\text{White Sedan} + \text{White SUV} + \text{White Truck} \\ &= 43 + 37 + 53 \\ &= 133 \end{aligned} \][/tex]
Step 4: Identify and count overlapping elements (white trucks).
Since we have already counted white trucks in both total trucks and total white vehicles, we need to subtract these from our total to avoid double-counting:
[tex]\[ \begin{aligned} &\text{White Trucks} = \text{White Truck} \\ &= 53 \end{aligned} \][/tex]
Step 5: Calculate the number of customers who purchased either a truck or a white vehicle.
To find the combined, we sum the total trucks and total white vehicles, then subtract the white trucks:
[tex]\[ \begin{aligned} &\text{Trucks or White} = \\ &\text{Total Trucks} + \text{Total White} - \text{White Trucks} \\ &= 79 + 133 - 53 \\ &= 159 \end{aligned} \][/tex]
Step 6: Calculate the probability that a randomly selected customer purchased a truck or a white vehicle.
Using the total number of purchases, the probability is:
[tex]\[ \begin{aligned} P(\text{Truck or White}) &= \frac{\text{Trucks or White}}{\text{Total Purchases}} \\ &= \frac{159}{219} \\ &\approx 0.7260 \quad \text{(approximation)} \end{aligned} \][/tex]
Therefore, the exact probability that a randomly selected customer purchased either a truck or a white vehicle is:
[tex]\[ P(\text{Truck or White}) = 0.726 \][/tex]
Step 1: Calculate the total number of purchases.
We will sum all the purchases for each type of car and color:
[tex]\[ \begin{aligned} &\text{Total purchases} = \\ &\text{Red Sedan} + \text{Red SUV} + \text{Red Truck} + \\ &\text{Blue Sedan} + \text{Blue SUV} + \text{Blue Truck} + \\ &\text{White Sedan} + \text{White SUV} + \text{White Truck} \\ &= 17 + 7 + 3 \\ &+ 17 + 19 + 23 \\ &+ 43 + 37 + 53 \\ &= 219 \\ \end{aligned} \][/tex]
Step 2: Calculate the total number of trucks.
Sum all the truck purchases for each color:
[tex]\[ \begin{aligned} &\text{Total Trucks} =\\ &\text{Red Truck} + \text{Blue Truck} + \text{White Truck}\\ &= 3 + 23 + 53 \\ &= 79 \end{aligned} \][/tex]
Step 3: Calculate the total number of white vehicles.
Sum all the purchases of white vehicles (Sedan, SUV, and Truck):
[tex]\[ \begin{aligned} &\text{Total White Vehicles} = \\ &\text{White Sedan} + \text{White SUV} + \text{White Truck} \\ &= 43 + 37 + 53 \\ &= 133 \end{aligned} \][/tex]
Step 4: Identify and count overlapping elements (white trucks).
Since we have already counted white trucks in both total trucks and total white vehicles, we need to subtract these from our total to avoid double-counting:
[tex]\[ \begin{aligned} &\text{White Trucks} = \text{White Truck} \\ &= 53 \end{aligned} \][/tex]
Step 5: Calculate the number of customers who purchased either a truck or a white vehicle.
To find the combined, we sum the total trucks and total white vehicles, then subtract the white trucks:
[tex]\[ \begin{aligned} &\text{Trucks or White} = \\ &\text{Total Trucks} + \text{Total White} - \text{White Trucks} \\ &= 79 + 133 - 53 \\ &= 159 \end{aligned} \][/tex]
Step 6: Calculate the probability that a randomly selected customer purchased a truck or a white vehicle.
Using the total number of purchases, the probability is:
[tex]\[ \begin{aligned} P(\text{Truck or White}) &= \frac{\text{Trucks or White}}{\text{Total Purchases}} \\ &= \frac{159}{219} \\ &\approx 0.7260 \quad \text{(approximation)} \end{aligned} \][/tex]
Therefore, the exact probability that a randomly selected customer purchased either a truck or a white vehicle is:
[tex]\[ P(\text{Truck or White}) = 0.726 \][/tex]
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