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

The problem presents an assertion and a reason, and we need to determine the relationship between them.

Let's analyze both the Assertion (A) and the Reason (R):

1. Assertion (A): If the probability of happening of an event is \(0.2p\), where \(p > 0\), then \(p\) can't be more than 5.

We know that probability values range from 0 to 1. Therefore, we must have:

• \(0 \leq 0.2p \leq 1\)

Solving for \(p\), we divide throughout by 0.2:

• \(0 \leq p \leq \frac{1}{0.2} = 5\)

This confirms that \(p\) cannot be more than 5. Therefore, Assertion (A) is true.

1. Reason (R): \(P(\bar{E}) = 1 - P(E)\) for an event \(E\).

This is a basic probability rule stating that the probability of the complement of an event is one minus the probability of the event happening. This rule is correct.

So, Reason (R) is also true.

However, Reason (R) does not explain the upper limit condition of \(p\) in the assertion. The assertion is not dependent on or explained by the reason. Therefore, we conclude:

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

Therefore, the correct answer is the option: "Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of Assertion (A)."