Let $A$ and $B$ be two events in a probability space with $P(A) = 0.3$, $P(B) = 0.5$, and $P(A \cap B) = 0.1$. Which of the following statements is/are TRUE?

We are given:

• $P(A) = 0.3$

• $P(B) = 0.5$

• $P(A \cap B) = 0.1$

Let us evaluate each statement:

1. Option A: For two events to be independent, $P(A \cap B) = P(A) \times P(B)$. Here, $P(A) \times P(B) = 0.3 \times 0.5 = 0.15 \neq 0.1$. Hence, $A$ and $B$ are not independent. *(Statement A is false)*

2. Option B: By the principle of inclusion-exclusion:

   $$P(A \cup B) = P(A) + P(B) - P(A \cap B) = 0.3 + 0.5 - 0.1 = 0.7$$

   *(Statement B is true)*

3. Option C: Using the complement rule for sets:

   $$P(A \cap B^c) = P(A) - P(A \cap B) = 0.3 - 0.1 = 0.2$$

   *(Statement C is true)*

4. Option D: By De Morgan's Law:

   $$P(A^c \cap B^c) = P((A \cup B)^c) = 1 - P(A \cup B) = 1 - 0.7 = 0.3 \neq 0.4$$

   *(Statement D is false)*

Thus, the correct options are B and C.