To find the probability that a randomly chosen customer purchased an SUV or a blue vehicle, we'll follow these steps:
### Step 1: Count the Total Number of Vehicles Purchased
First, we need to calculate the total number of vehicles purchased across all categories.
- Red Sedan: 17
- Red SUV: 7
- Red Truck: 3
- Blue Sedan: 17
- Blue SUV: 19
- Blue Truck: 23
- White Sedan: 43
- White SUV: 37
- White Truck: 53
Adding them up, we get:
[tex]\[ 17 + 7 + 3 + 17 + 19 + 23 + 43 + 37 + 53 = 219 \][/tex]
So, the total number of vehicles purchased is 219.
### Step 2: Count the Number of SUVs
Next, we need to determine the total number of SUVs purchased.
- Red SUV: 7
- Blue SUV: 19
- White SUV: 37
Adding these, we get:
[tex]\[ 7 + 19 + 37 = 63 \][/tex]
So, 63 SUVs were purchased.
### Step 3: Count the Number of Blue Vehicles
Then, we calculate the total number of blue vehicles purchased.
- Blue Sedan: 17
- Blue SUV: 19
- Blue Truck: 23
Adding these, we get:
[tex]\[ 17 + 19 + 23 = 59 \][/tex]
So, 59 blue vehicles were purchased.
### Step 4: Count the Number of Blue SUVs
Now, we identify how many of the SUVs are also blue. This has already been counted in both the number of SUVs and the number of blue vehicles.
- Blue SUV: 19
Therefore, there are 19 blue SUVs.
### Step 5: Calculate the Probability of SUV or Blue Vehicle
Using the principle of inclusion and exclusion, we find the number of customers who purchased an SUV or a blue vehicle:
[tex]\[ \text{Number of SUV or Blue} = \text{Total SUVs} + \text{Total Blue} - \text{Blue SUVs} \][/tex]
[tex]\[ = 63 + 59 - 19 = 103 \][/tex]
The probability is then the ratio of this quantity to the total number of vehicles purchased:
[tex]\[ P(\text{SUV or Blue}) = \frac{103}{219} \][/tex]
### Conclusion
Thus, the probability that a randomly chosen customer purchased an SUV or a blue vehicle is:
[tex]\[ P(\text{SUV or Blue}) = \frac{103}{219} \][/tex]
This fraction cannot be simplified further, so the probability remains [tex]\(\frac{103}{219}\)[/tex]. However, as a decimal, it is approximately [tex]\(0.4703196347031963\)[/tex], which corresponds to 47.03%.
