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At a carwash, the purchases for one week are recorded in the table below:

\begin{tabular}{|l|c|c|c|}
\hline
& Rinse & Wax & \begin{tabular}{c}
Rinse and \\
Wax
\end{tabular} \\
\hline
SUV & 11 & 7 & 13 \\
\hline
Sedan & 31 & 19 & 17 \\
\hline
Van & 41 & 29 & 23 \\
\hline
\end{tabular}

If we choose a customer at random, what is the probability that they have purchased a job for a sedan or a wax?

[tex]\[ P (\text{Sedan or Wax}) = \underline{[?]} \][/tex]

Give your answer in simplest form.


Sagot :

To determine the probability that a randomly chosen customer purchased a job for a sedan or a wax, we need to follow these steps:

1. Calculate the total number of customers:
We first sum the number of purchases across all vehicle types and services.
[tex]\[ \text{Total Customers} = 11 (\text{SUV rinse}) + 7 (\text{SUV wax}) + 13 (\text{SUV rinse and wax}) + 31 (\text{Sedan rinse}) + 19 (\text{Sedan wax}) + 17 (\text{Sedan rinse and wax}) + 41 (\text{Van rinse}) + 29 (\text{Van wax}) + 23 (\text{Van rinse and wax}) = 191 \][/tex]

2. Calculate the number of customers who have purchased a job for a sedan:
Add up the number of customers who purchased any type of service for a sedan.
[tex]\[ \text{Sedan Customers} = 31 (\text{Sedan rinse}) + 19 (\text{Sedan wax}) + 17 (\text{Sedan rinse and wax}) = 67 \][/tex]

3. Calculate the number of customers who have purchased a wax job:
Sum the number of customers who purchased any type of waxing service.
[tex]\[ \text{Wax Customers} = 7 (\text{SUV wax}) + 19 (\text{Sedan wax}) + 29 (\text{Van wax}) = 55 \][/tex]

4. Calculate the number of customers who purchased both a job for a sedan and a wax job:
Notice here that the customers who purchased "Sedan wax" have been counted in both Sedan Customers and Wax Customers.
[tex]\[ \text{Sedan and Wax Customers} = 19 \][/tex]

5. Use the principle of inclusion and exclusion to calculate the number of customers who have either purchased a job for a sedan or a wax job:
We add the number of sedan customers and the number of wax customers, and subtract the double-counted customers who fall into both categories.
[tex]\[ \text{Sedan or Wax Customers} = \text{Sedan Customers} + \text{Wax Customers} - \text{Sedan and Wax Customers} \][/tex]
Substituting the values we calculated:
[tex]\[ \text{Sedan or Wax Customers} = 67 + 55 - 19 = 103 \][/tex]

6. Calculate the probability:
[tex]\[ P(\text{Sedan or Wax}) = \frac{\text{Number of Sedan or Wax Customers}}{\text{Total Customers}} = \frac{103}{191} \approx 0.539 \][/tex]

Thus, the probability that a randomly chosen customer has purchased a job for a sedan or a wax job is approximately [tex]\( 0.539 \)[/tex] or [tex]\(\frac{103}{191}\)[/tex] in simplest form.
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