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Let's break down the problem step by step.

We are given:

- 37 vehicles total.
  - 18 are sedans.
  - 9 are trucks.
  - 10 are vans.

The task is to find the probability that the first vehicle selected is a truck, and the second vehicle selected is a van, assuming both are selected randomly.

### Step 1: Probability of selecting the first truck
There are 9 trucks out of 37 vehicles, so the probability of selecting a truck first is:
\[
P(\text{Truck first}) = \frac{9}{37}
\]

### Step 2: Probability of selecting the second van (after the truck is removed)
After the first truck is selected, there are now 36 vehicles left, and 10 of them are vans. So, the probability of selecting a van next is:
\[
P(\text{Van second | Truck first}) = \frac{10}{36}
\]

### Step 3: Multiply the probabilities
The combined probability of both events (first truck, then van) is:
\[
P(\text{Truck first and Van second}) = \frac{9}{37} \times \frac{10}{36}
\]
\[
P = \frac{9 \times 10}{37 \times 36} = \frac{90}{1332}
\]

### Step 4: Simplify and calculate
Now, let's calculate the final probability:
\[
P = \frac{90}{1332} \approx 0.068
\]

Thus, the probability that the first vehicle is a truck and the second is a van is approximately **0.068**.

Would you like more details or have any questions?

### Related Questions:
1. What is the probability that both selected vehicles are trucks?
2. How would the probability change if the first vehicle selected is a van?
3. What is the probability that neither vehicle selected is a sedan?
4. What if three vehicles are selected? How would the probability be calculated for specific sequences?
5. How does the probability change if trucks and vans are equally likely?

### Tip:
When dealing with probability problems involving sequential selections, always adjust the total number of items and the number of specific items left after each step.

An automobile manufacturing plant produced 37 vehicles today: 18 were sedans, 9 were trucks, and 10 were vans. Plant managers are going to select two of these vehicles for a thorough inspection. The first vehicle will be selected at random, and the second vehicle will be selected at random from the remaining vehicles. What is the probability that the first vehicle selected is a truck and the second vehicle is a van?

Learn how to calculate the probability of selecting a truck followed by a van from 37 vehicles (18 sedans, 9 trucks, 10 vans). This detailed solution explains the use of probability rules and combinatorics, suitable for Grades 9-11 students.

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