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.
