Question

Directions: In Question Numbers 19 and 20, a statement of Assertion (A) is followed by a statement of Reason (R).
Choose the correct option from the following:
(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of 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.

In an experiment of throwing a die,
Assertion (A): Event $E_1$: getting a number less than 3 and Event $E_2$: getting a number greater than 3 are complementary events.
Reason (R): If two events $E$ and $F$ are complementary events, then $P(E) + P(F) = 1$.

• A. Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of 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:

Complementary events together cover all possible outcomes with no overlap or omission.

Answer 1

The Correct Option is D

Step 1: Sample space for a die roll:
S = {1, 2, 3, 4, 5, 6}

Step 2: Event definitions
E₁: Number less than 3 = {1, 2}
E₂: Number greater than 3 = {4, 5, 6}

Step 3: Are E₁ and E₂ complementary?
Complementary events require:
(a) Mutually exclusive → E₁ ∩ E₂ = ∅ ⇒ True
(b) Collectively exhaustive → E₁ ∪ E₂ covers the entire sample space
E₁ ∪ E₂ = {1, 2, 4, 5, 6} (missing 3) ⇒ Not exhaustive

Conclusion for Step 3: E₁ and E₂ are not complementary.

Step 4: Reason Analysis
The reason states: "If events E and F are complementary, then P(E) + P(F) = 1"
This is a true mathematical identity for complementary events.

Final Conclusion:
Assertion (A) is false
Reason (R) is true