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: Thinking through a real-life experiment. 
Consider two separate experiments, like tossing a coin and rolling a die.
The result of the coin toss does not affect the outcome of the die roll.
Such events are called independent events.

Step 2: Probability reasoning for independent actions.
When two actions are independent, the chance of both happening together depends only on how likely each one is individually.
So, the probability of both events occurring together is found by combining their individual probabilities.

Step 3: Mathematical expression.
If event \( A \) occurs with probability \( P(A) \) and event \( B \) occurs with probability \( P(B) \), then the probability that both occur together is:
\[ P(A \cap B) = P(A) \times P(B) \]
This rule holds only because the events do not influence each other.

Step 4: Verifying with conditional probability.
For independent events, the occurrence of \( B \) does not change the chance of \( A \):
\[ P(A|B) = P(A) \]
Substituting into the conditional probability formula confirms the same result for \( P(A \cap B) \).

Step 5: Eliminating incorrect choices.
(A) Addition rules apply to mutually exclusive events, not independent ones.
(C) and (D) do not represent the correct relationship for independent events.
(B) Correctly states the multiplication rule for independent events.

Final Conclusion:
For two independent events, the probability that both occur is given by:
\[ \boxed{P(A \cap B) = P(A)P(B)} \]