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 when a die is thrown:
S = {1, 2, 3, 4, 5, 6}

Step 2: Define the events
E₁: Getting a number less than 3 = {1, 2}
E₂: Getting a number greater than 3 = {4, 5, 6}

Step 3: Check if E₁ and E₂ are complementary
Complementary events must satisfy two conditions:
(a) Mutually exclusive → E₁ ∩ E₂ = ∅ ⇒ True
(b) Collectively exhaustive → E₁ ∪ E₂ should cover the whole sample space
E₁ ∪ E₂ = {1, 2, 4, 5, 6} which is missing 3 ⇒ Not exhaustive

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

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

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