Identify multiples of 4

In the range 1 to 30, multiples of 4 are $4, 8, 12, 16, 20, 24, 28$. The count is $n(A) = 7$.

$n(A) = 7$

Identify multiples of 6

In the range 1 to 30, multiples of 6 are $6, 12, 18, 24, 30$. The count is $n(B) = 5$.

$n(B) = 5$

Identify intersection

Multiples of both 4 and 6 are multiples of their LCM, which is 12. These are $12, 24$. The count is $n(A \cap B) = 2$.

$n(A \cap B) = 2$

Apply inclusion-exclusion principle

Calculate the total count of numbers that are multiples of 4 or 6 using $n(A \cup B) = n(A) + n(B) - n(A \cap B) = 7 + 5 - 2 = 10$.

$n(A \cup B) = 10$

Calculate probability

The probability is the ratio of the favorable outcomes to the total outcomes (30). $P = 10 / 30 = 1 / 3$ is incorrect per the options; re-checking the count: 4, 8, 12, 16, 20, 24, 28 (7), 6, 12, 18, 24, 30 (5). Union = {4, 6, 8, 12, 16, 18, 20, 24, 28, 30} = 10 elements. $10/30 = 1/3$. Wait, checking the provided answer key C (3/10). Re-counting: 4, 8, 12, 16, 20, 24, 28 (7). 6, 12, 18, 24, 30 (5). Union count is 10. The probability is $10/30 = 1/3$.

$P = \frac{10}{30} = \frac{1}{3}$