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This problem involves conditional probability, where we need to calculate the probability that a victim of a deadly side-impact accident was driving an SUV, given certain data.

### Data provided:
- 27.7% of all light vehicles were **light trucks**.
- 27.7% of all light vehicles were **SUVs**.
- 44.6% of all light vehicles were **cars**.

The probabilities of a severe side-impact crash proving deadly for each vehicle type are:
- **Light Truck**: 0.210
- **SUV**: 0.371
- **Car**: 1.000

### Step-by-Step Solution:

We need to calculate the probability that the victim was driving an **SUV** given that the crash was deadly. This can be solved using **Bayes' Theorem**.

Bayes' Theorem formula:
\[
P(\text{SUV} | \text{Deadly}) = \frac{P(\text{Deadly} | \text{SUV}) \cdot P(\text{SUV})}{P(\text{Deadly})}
\]
Where:
- \( P(\text{SUV} | \text{Deadly}) \) = Probability that the vehicle is an SUV, given that the crash was deadly.
- \( P(\text{Deadly} | \text{SUV}) = 0.371 \) = Probability that a crash is deadly, given that the vehicle is an SUV.
- \( P(\text{SUV}) = 0.277 \) = Probability that a vehicle is an SUV.
- \( P(\text{Deadly}) \) = Total probability of a deadly crash.

To calculate \( P(\text{Deadly}) \), we use the law of total probability:
\[
P(\text{Deadly}) = P(\text{Deadly} | \text{Light Truck}) \cdot P(\text{Light Truck}) + P(\text{Deadly} | \text{SUV}) \cdot P(\text{SUV}) + P(\text{Deadly} | \text{Car}) \cdot P(\text{Car})
\]
Substitute the values:
\[
P(\text{Deadly}) = (0.210 \cdot 0.277) + (0.371 \cdot 0.277) + (1.000 \cdot 0.446)
\]
Now I will calculate this.The total probability of a deadly crash, \( P(\text{Deadly}) \), is approximately 0.6069.

Now, we can use Bayes' Theorem to calculate the probability that the victim was driving an **SUV** given that the crash was deadly:
\[
P(\text{SUV} | \text{Deadly}) = \frac{P(\text{Deadly} | \text{SUV}) \cdot P(\text{SUV})}{P(\text{Deadly})}
\]
Substituting the values:
\[
P(\text{SUV} | \text{Deadly}) = \frac{0.371 \cdot 0.277}{0.6069}
\]
I will now compute this.The probability that the victim of a deadly side-impact accident was driving an **SUV** is approximately **0.1693** or **16.93%**.

Would you like further clarification or details on any part of the process?

Here are some related questions to expand your understanding:
1. How would the probability change if the likelihood of a deadly crash for cars was lower than 1.000?
2. Can Bayes' Theorem be applied to other conditional probability problems involving different types of accidents?
3. How would you approach the problem if there was missing data for one type of vehicle?
4. What would happen if the distribution of vehicle types changed over time?
5. How would the result differ if the crash probabilities were reported as a function of road conditions?

**Tip:** When solving conditional probability problems, clearly identify the "given" condition and break down the formula to avoid confusion.

What is the probability that the victim of a deadly side-impact accident was driving an SUV?

This problem explores the probability that a victim of a deadly side-impact accident was driving an SUV. Using Bayes' Theorem and the Law of Total Probability, the problem provides step-by-step calculations to solve this real-world scenario. Suitable for students in Grades 11-12.

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