Question

Probability of occurrence of an event A is 1/2 and that of B is 3/10. If A and B are mutually exclusive, then the probability of occurrence of neither A nor B is

• A. $\frac{4}{5}$
• B. $\frac{3}{5}$
• C. $\frac{2}{5}$
• D. $\frac{1}{5}$

Hint:

For mutually exclusive events, think of probability as just adding areas. If A covers 50% of the space and B covers 30% (with no overlap), together they cover 80%. What's left over ("neither") is simply $100% - 80% = 20%$, which is $1/5$.

Answer 1

The Correct Option is D

Step 1: Given Data
\[ P(A) = \frac{1}{2}, P(B) = \frac{3}{10} \] Since $A$ and $B$ are mutually exclusive: \[ P(A \cap B) = 0 \]
Step 2: Find $P(A \cup B)$
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \] \[ P(A \cup B) = \frac{1}{2} + \frac{3}{10} \] \[ = \frac{5}{10} + \frac{3}{10} = \frac{8}{10} = \frac{4}{5} \]
Step 3: Probability of Neither A nor B
\[ P(\text{neither A nor B}) = 1 - P(A \cup B) \] \[ = 1 - \frac{4}{5} = \frac{1}{5} \]
Step 4: Final Answer
\[ \boxed{\frac{1}{5}} \]