To find the probability that a number chosen at random from the integers 11 to 20 (inclusive) is either prime or a multiple of 3, follow these steps:

1. List the numbers from 11 to 20:
\[ 11, 12, 13, 14, 15, 16, 17, 18, 19, 20 \]

2. Identify the prime numbers in this range:
Prime numbers: \(11, 13, 17, 19\)

3. Identify the multiples of 3 in this range:
Multiples of 3: \(12, 15, 18\)

4. Combine these sets and eliminate duplicates:
Numbers that are either prime or multiples of 3:
\[ 11, 12, 13, 15, 17, 18, 19 \]

5. Count the total unique numbers:
There are 7 unique numbers: \(11, 12, 13, 15, 17, 18, 19\)

6. Count the total number of integers from 11 to 20:
Total integers: 10

7. Calculate the probability:
\[
\text{Probability} = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{7}{10}
\]

Therefore, the correct answer is:

A. \(\frac{7}{10}\)