Question

If $A$ and $B$ are independent events with $P(A)=0.3$ and $P(B)=0.4$, then $P(A \cap B)$ is

• A. $1.2$
• B. $0.12$
• C. $0.7$
• D. $\frac{3}{4}$

Hint:

For independent events always remember the rule: $P(A \cap B) = P(A)\times P(B)$.

Answer 1

The Correct Option is B

Step 1: Recall the definition of independent events.
Two events $A$ and $B$ are said to be independent if the occurrence of one event does not affect the probability of the other event. Mathematically, for independent events we have \[ P(A \cap B) = P(A) \times P(B) \]
Step 2: Substitute the given probabilities.
We are given \[ P(A)=0.3 \] \[ P(B)=0.4 \] Using the formula for independent events \[ P(A \cap B) = P(A)\times P(B) \]
Step 3: Perform the multiplication.
\[ P(A \cap B) = 0.3 \times 0.4 \] \[ P(A \cap B) = 0.12 \]
Step 4: Verify the result.
Since probabilities always lie between $0$ and $1$, the value $0.12$ is a valid probability.

Step 5: Conclusion.
Therefore the probability that both events $A$ and $B$ occur together is \[ P(A \cap B)=0.12 \] Final Answer: $\boxed{0.12}$