Question

Assertion (A) : If probability of happening of an event is \(0.2p\), \(p>0\), then \(p\) can't be more than 5.
Reason (R) : \(P(\bar{E}) = 1 - P(E)\) for an event \(E\).

• A. Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
• B. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of Assertion (A).
• C. Assertion (A) is true, but Reason (R) is false.
• D. Assertion (A) is false, but Reason (R) is true.

Hint:

Probability questions often hide constraints. Remember the two main bounds: \(P(E) \geq 0\) and \(P(E) \leq 1\). Most "find the range of variable" problems in probability rely on these.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
Probability of any event \(E\) always lies between 0 and 1 inclusive (\(0 \leq P(E) \leq 1\)).
The sum of probability of occurrence and non-occurrence of an event is 1.
Step 3: Detailed Explanation:
Assertion (A): Given \(P(E) = 0.2p\).
Since \(P(E) \leq 1\):
\[ 0.2p \leq 1 \]
\[ p \leq \frac{1}{0.2} \]
\[ p \leq 5 \]
So, \(p\) cannot be more than 5. (A) is true.
Reason (R): It is a fundamental property of probability that \(P(E) + P(\bar{E}) = 1\), so \(P(\bar{E}) = 1 - P(E)\). (R) is true.
Connection: While both are true, the reason for \(p \leq 5\) is the definition of the range of probability (\(0 \leq P(E) \leq 1\)), not specifically the formula for the complement event. Thus, (R) is not the explanation for (A).
Step 4: Final Answer: Both (A) and (R) are true, but (R) is not the correct explanation of (A).