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. $\frac{7}{10}$