The fastest way to solve these type of question is with the help of Venn Diagram.
For this question, Venn Diagram will look like this :
P(A) = 0.3, P(B) = 0.5, P(A $\cap$ B) = 0.1
Let's try to observe every option one by one.
Option A : The two events 𝐴 and 𝐵 are independent
Two events are said to be Independent when : P(A $\cap$ B) = P(A).P(B)
P(A $\cap$ B) = 0.1
P(A).P(B) = 0.3 x 0.5 = 0.15
P(A $\cap$ B) $\neq$ P(A).P(B)
So, Option A is Incorrect.
Option B : P(A $\cup$ B) = 0.7.
P(A $\cup$ B) = P(A) + P(B) - P(A $\cap$ B)
= 0.3 + 0.5 - 0.1 = 0.7
So, Option B is Correct.
Option C : P($A \cap B^{c}$) = 0.2
P($A \cap B^{c}$) = P(A-B) = P(Only A) = 0.2
From Above Venn diagram it is clear that this is True.
So, Option C is Correct.
Option D : P($A^{c} \cap B^{c}$) = 0.4
P($A^{c} \cap B^{c}$) = P($(A \cup B)^{c}$) = 1 - P(A $\cup$ B)
From venn diagram it is clear that this value is 0.3 not 0.4.
This statement is False.
So, Option D is Incorrect.
Correct Answer : B, C