Question

Given below are two statements, one is labelled as Assertion (A) and other one labelled as Reason (R).
Assertion (A) : The probability of a sure event is 1.
Reason (R) : Let E be an event. Then \(0 \le P(E) \le 1\).
In light of the above statements, choose the correct answer from the options given below:

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

Hint:

To verify the relationship, read them together using "because":
"The probability of a sure event is 1 because any event's probability is between 0 and 1."
This is not a logical cause-and-effect relationship, confirming that (R) is not the correct explanation of (A).

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
According to Kolmogorov's axioms of probability, the probability of any event \(E\) is a real number bounded between 0 and 1.
A sure event is an event that is guaranteed to happen, which is represented by the entire sample space \(S\).

Step 2: Detailed Explanation:
Let us evaluate the Assertion and the Reason:
- Assertion (A): "The probability of a sure event is 1."
This is a true statement. According to the second axiom of probability:
\[ P(S) = 1 \]
Where \(S\) is the sample space. Since a sure event contains all possible outcomes, its probability is exactly 1.
- Reason (R): "Let E be an event. Then \(0 \le P(E) \le 1\)."
This is also a true statement. It represents the first axiom of probability (non-negativity) and the consequence that no probability can exceed the probability of the sample space.
Let us evaluate the relationship:
While both (A) and (R) are true mathematical facts, the Reason (R) is not the correct explanation of (A).
The reason why the probability of a sure event is 1 is because the sure event is equal to the sample space \(S\), and by axiom, \(P(S) = 1\).
The inequality \(0 \le P(E) \le 1\) is a general property of all events, but it does not specifically explain why the upper bound must be achieved by a sure event.
Therefore, both (A) and (R) are true, but (R) is not the correct explanation of (A).

Step 3: Final Answer:
The correct option is (B).