If $P(A) = \dfrac{1}{2}$, $P(B) = \dfrac{1}{3}$ and $A$ and $B$ are independent events, then $P(A \cap B)$ is:

Step-by-step Solution:

For independent events $A$ and $B$:

$$P(A \cap B) = P(A) \times P(B)$$

$$= \frac{1}{2} \times \frac{1}{3}$$

$$= \boxed{\frac{1}{6}}$$

Key Concept: Two events $A$ and $B$ are independent if the occurrence of one does not affect the probability of the other.

Hence, Option B is correct.