Introduction
In probability theory, determining whether two events $A$ and $B$ are independent is a fundamental skill. Events are considered independent if the occurrence of one does not affect the probability of the other occurring.
The Mathematical Definition
Two events $A$ and $B$ are independent if and only if: $$P(A \cap B) = P(A) \times P(B)$$
Step-by-Step Problem Solving
Given:
- $P(A) = 0.4$
- $P(B) = 0.3$
- $P(A \cup B) = 0.56$
Step 1: Find the intersection $P(A \cap B)$
We use the Inclusion-Exclusion Principle: $$P(A \cup B) = P(A) + P(B) - P(A \cap B)$$
Substitute the known values: $$0.56 = 0.4 + 0.3 - P(A \cap B)$$ $$0.56 = 0.7 - P(A \cap B)$$ $$P(A \cap B) = 0.7 - 0.56 = 0.14$$
Step 2: Check the independence condition
Calculate $P(A) \times P(B)$: $$0.4 \times 0.3 = 0.12$$
Compare with $P(A \cap B)$: $$0.14 \neq 0.12$$
Conclusion
Since $P(A \cap B) \neq P(A) \times P(B)$, the events $A$ and $B$ are not independent.