Question

If two events $A$ and $B$ are independent, then:

• A. $P(A \cap B) = P(A) + P(B)$
• B. $P(A \cap B) = P(A)P(B)$
• C. $P(A|B) = P(A) + P(B)$
• D. $P(A \cup B) = P(A)P(B)$

Hint:

Independence in probability always leads to {multiplication of probabilities}, never addition.

Answer 1

The Correct Option is B

Step 1: Meaning of independent events.
Two events $A$ and $B$ are called independent if the occurrence of one event does not influence the occurrence of the other. In simple terms, knowing whether $A$ has occurred gives no information about $B$, and vice versa.
Step 2: Conditional probability viewpoint.
By definition, if $A$ and $B$ are independent, then \[ P(A|B) = P(A) \quad \text{and} \quad P(B|A) = P(B) \] This means the probability of $A$ remains the same even after $B$ has occurred.
Step 3: Deriving the required formula.
We know the general formula of conditional probability: \[ P(A|B) = \frac{P(A \cap B)}{P(B)} \] Since $P(A|B) = P(A)$ for independent events, we get: \[ \frac{P(A \cap B)}{P(B)} = P(A) \] Multiplying both sides by $P(B)$, \[ P(A \cap B) = P(A)P(B) \]
Step 4: Checking the given options.
(A) Incorrect — addition rule applies to mutually exclusive events, not independent ones.
(B) Correct — this is the defining condition of independent events.
(C) Incorrect — for independent events, $P(A|B) = P(A)$, not a sum.
(D) Incorrect — probability of union follows a different formula.
Step 5: Final conclusion.
If events $A$ and $B$ are independent, then the probability of their intersection is \[ \boxed{P(A \cap B) = P(A)P(B)} \]