A company manufactures scooters at two plants, A and B. Plant A produces 80% and plant B produces 20% of the total product. 85% of the scooters produced at plant A and 65% of the scooters produced at plant B are of standard quality. A scooter produced by the company is selected at random and it is found to be of standard quality. What is the probability that it was manufactured at plant A?

Explanation

To solve the problem, we will use Bayes' Theorem, which allows us to find the probability of an event based on prior knowledge of conditions related to the event.

### Step-by-Step Solution:

1. **Define the Events**:
- Let \( A \) be the event that a scooter is manufactured at Plant A.
- Let \( B \) be the event that a scooter is manufactured at Plant B.
- Let \( S \) be the event that a scooter is of standard quality.

2. **Given Probabilities**:
- \( P(A) = 0.8 \) (80% of scooters are from Plant A)
- \( P(B) = 0.2 \) (20% of scooters are from Plant B)
- \( P(S|A) = 0.85 \) (85% of scooters from Plant A are of standard quality)
- \( P(S|B) = 0.65 \) (65% of scooters from Plant B are of standard quality)

3. **Find \( P(S) \)**:
- We need to calculate the total probability of selecting a standard quality scooter, \( P(S) \):
\[ P(S) = P(S|A) \cdot P(A) + P(S|B) \cdot P(B) \]
Substituting the values:
\[ P(S) = (0.85 \cdot 0.8) + (0.65 \cdot 0.2) \]
\[ P(S) = 0.68 + 0.13 = 0.81 \]

4. **Apply Bayes' Theorem**:
- We want to find \( P(A|S) \), the probability that a scooter is from Plant A given that it is of standard quality:
\[ P(A|S) = \frac{P(S|A) \cdot P(A)}{P(S)} \]
Substituting the known values:
\[ P(A|S) = \frac{0.85 \cdot 0.8}{0.81} \]
\[ P(A|S) = \frac{0.68}{0.81} \]

5. **Calculate the Final Probability**:
- Now we calculate \( \frac{0.68}{0.81} \):
\[ P(A|S) \approx 0.8395 \]

### Conclusion:
The probability that a randomly selected scooter of standard quality was manufactured at Plant A is approximately \( 0.8395 \) or \( 83.95\% \).