An automobile manufacturing plant produced  vehicles today:  were motorcycles,  were sedans, and  were vans. (Each vehicle falls into only one of these categories.) Plant managers are going to select two of these vehicles for a thorough inspection. The first vehicle will be selected at random, and then the second vehicle will be selected at random from the remaining vehicles. What is the probability that two sedans will be selected?
Do not round your intermediate computations. Round your final answer to three decimal places.

(If necessary, consult a list of formulas.)

To find the probability that two sedans are selected in a row, we'll break down the problem step by step.

Let's denote:
- The total number of vehicles produced as \( T \).
- The number of motorcycles as \( M \).
- The number of sedans as \( S \).
- The number of vans as \( V \).

Given that the plant produced a total of \( T \) vehicles, with \( S \) sedans, the probability of selecting a sedan on the first draw is:

\[
P(\text{first sedan}) = \frac{S}{T}
\]

After one sedan is selected, there are \( S - 1 \) sedans remaining out of a total of \( T - 1 \) vehicles. The probability of selecting a sedan on the second draw, given that the first one was a sedan, is:

\[
P(\text{second sedan} \mid \text{first sedan}) = \frac{S-1}{T-1}
\]

The probability of both events (selecting two sedans in a row) occurring is the product of the two probabilities:

\[
P(\text{two sedans}) = P(\text{first sedan}) \times P(\text{second sedan} \mid \text{first sedan}) = \frac{S}{T} \times \frac{S-1}{T-1}
\]

Finally, let's substitute the given values. Since the problem states specific numbers (though the values aren't visible here), assume we have them. Let’s denote:

- \( S = \) the number of sedans
- \( T = S + M + V \) the total number of vehicles

The probability calculation becomes:

\[
P(\text{two sedans}) = \frac{S}{T} \times \frac{S-1}{T-1}
\]

Now let's calculate it:

1. Plug in the specific values for \( S \) and \( T \).
2. Calculate the result and round it to three decimal places.

**Example Calculation (assuming hypothetical numbers):**

If \( S = 20 \), \( T = 50 \):

\[
P(\text{two sedans}) = \frac{20}{50} \times \frac{19}{49} = \frac{20 \times 19}{50 \times 49} = \frac{380}{2450} \approx 0.155
\]

So, the probability would be approximately **0.155** (rounded to three decimal places).

Let me know the specific numbers, and I can compute the exact probability for you.

Do you want more details or have any questions? 

Here are some related questions:
1. What is the probability of selecting two motorcycles in a row?
2. How would the probability change if the total number of sedans were increased?
3. What is the probability of selecting one sedan and one van?
4. How does the probability change if three vehicles are selected instead of two?
5. What is the probability of not selecting a sedan at all?

**Tip:** When calculating probabilities involving multiple steps, it's crucial to consider how each step affects the total number of possible outcomes.

Learn how to calculate the probability of selecting two sedans in a manufacturing plant where vehicles are categorized as motorcycles, sedans, and vans. Understand the step-by-step calculation using the formula for probability of independent events. Suitable for high school students and anyone interested in probability problems.

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