To determine the probability that a randomly chosen customer purchased a "Rinse and Wax" for a van, follow these detailed steps:
### Step 1: Identify Relevant Data
From the given table, collect the total purchases for "Rinse and Wax" across different vehicle types:
- For SUVs: 13 purchases
- For Sedans: 17 purchases
- For Vans: 23 purchases
### Step 2: Compute the Total Number of Customers
Total customers for each service per vehicle type must include "Rinse," "Wax," and "Rinse and Wax."
- Total SUV customers = 11 (Rinse) + 7 (Wax) + 13 (Rinse and Wax) = 31 customers
- Total Sedan customers = 31 (Rinse) + 19 (Wax) + 17 (Rinse and Wax) = 67 customers
- Total Van customers = 41 (Rinse) + 29 (Wax) + 23 (Rinse and Wax) = 93 customers
Sum these up to get the total number of customers:
[tex]\[ \text{Total customers} = 31 (\text{SUV}) + 67 (\text{Sedan}) + 93 (\text{Van}) = 191 \text{ customers} \][/tex]
### Step 3: Identify the Number of Relevant Customers
The relevant customers are those who purchased a "Rinse and Wax" for a van. According to the data, this number is:
[tex]\[ \text{Rinse and Wax for Van} = 23 \text{ customers} \][/tex]
### Step 4: Compute the Probability
The probability [tex]\( P \)[/tex] of choosing a customer who purchased a "Rinse and Wax" for a van is the ratio of the number of van "Rinse and Wax" purchases to the total number of customers.
[tex]\[ P(\text{Van and Rinse & Wax}) = \frac{\text{Number of Rinse and Wax for Van}}{\text{Total number of customers}} = \frac{23}{191} \][/tex]
### Final Probability Calculation:
[tex]\[ P(\text{Van and Rinse & Wax}) = \frac{23}{191} \][/tex]
Therefore, the probability that a randomly selected customer purchased a "Rinse and Wax" for a van is:
[tex]\[ \boxed{\frac{23}{191}} \][/tex]
This fraction is already in its simplest form.
