Question

Assertion (A) : If probability of happening of an event is 0.2p, p $\gt $ 0, then p can’t be more than 5.
Reason (R) : P( \(\overline{E}\) ) = 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:

To easily check if the Reason is the correct explanation for the Assertion, read the Assertion statement, insert the word "BECAUSE", and then read the Reason statement:
"If the probability of an event is \(0.2p\), then \(p \le 5\) BECAUSE the probability of the complementary event is \(1 - P(E)\)."
This combined statement does not make logical sense, showing that the Reason is not the correct explanation!

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
The topic of this question is Probability.
This is an Assertion-Reason style question where we need to independently evaluate the mathematical truth of two statements: the Assertion (A) and the Reason (R).
If both statements are true, we must then determine whether the Reason (R) provides the correct logical explanation for the Assertion (A).

Step 2: Key Formula or Approach:
1. The probability of any event \(E\), denoted by \(P(E)\), is bounded. It must lie between 0 and 1:
\[ 0 \le P(E) \le 1 \] 2. The sum of the probability of an event and its complement is 1:
\[ P(E) + P(\overline{E}) = 1 \implies P(\overline{E}) = 1 - P(E) \]

Step 3: Detailed Explanation:

• Analyze Assertion (A):
Let the probability of the event be \(P(E) = 0.2p\), where \(p \gt 0\).
Since the probability of any event cannot exceed 1:
\[ P(E) \le 1 \] \[ 0.2p \le 1 \implies p \le \frac{1}{0.2} \implies p \le 5 \] Thus, \(p\) cannot be more than 5. Assertion (A) is true.

• Analyze Reason (R):
The statement \(P(\overline{E}) = 1 - P(E)\) is a standard mathematical relationship for complementary events. Thus, Reason (R) is true.

• Check if Reason (R) explains Assertion (A):
The Assertion is proved using the inequality \(P(E) \le 1\).
It does not utilize the complementary probability formula.
Therefore, Reason (R) is not the correct explanation for Assertion (A).


Step 4: Final Answer:
Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of Assertion (A).
Therefore, the correct option is (B).