Let A and B be two events in a probability space with P(A) = 0.3, P(B) = 0.5, and P(A B) = 0.1. Which of the following statements is/are TRUE? A) The two events A and B are independent B) P(A B) = 0.7 C) P(A Bc) = 0.2, where Bc is the complement of the event B D) P(Ac Bc) = 0.4, where Ac and Bc are the complements of the events A and B, respectively

Official answer: B) P(A B) = 0.7; C) P(A Bc) = 0.2, where Bc is the complement of the event B. Explanation: We are given: - P(A) = 0.3 - P(B) = 0.5 - P(A B) = 0.1 Let us evaluate each statement: 1. Option A: For two events to be independent, P(A B) = P(A) P(B). Here, P(A) P(B) = 0.3 0.5 = 0.15 0.1. Hence, A and B are not independent. *(Statement A is false)* 2. Option B: By the principle of inclusion-exclusion: P(A B) = P(A) + P(B) - P(A B) = 0.3 + 0.5 - 0.1 = 0.7 *(Statement B is true)* 3. Option C: Using the complement rule for sets: P(A Bc) = P(A) - P(A B) = 0.3 - 0.1 = 0.2 *(Statement C is true)* 4. Option D: By De Morgan's Law: P(Ac Bc) = P((A B)c) = 1 - P(A B) = 1 - 0.7 = 0.3 0.4 *(Statement D is false)* Thus, the correct options are B and C.

A) The two events A and B are independent

C) P(A Bc) = 0.2, where Bc is the complement of the event B

D) P(Ac Bc) = 0.4, where Ac and Bc are the complements of the events A and B, respectively

Engineering Mathematics · Probability and Statistics

Official IIT answer key · IISc Bangalore · Audited Aug 2026

Take topic test

GATE CSE 2024 Set 1 Q27 · MSQ · 1 mark

Let $A$ and $B$ be two events in a probability space with $P(A) = 0.3$ , $P(B) = 0.5$ , and $P(A \cap B) = 0.1$ . Which of the following statements is/are TRUE?

Select one or more options

Key concept

Probability and Statistics

Step-by-step verified solution

Examiner trap & misconception

Topic notes

How GATE tests Probability and Statistics

Probability and Statistics in GATE CS is consistently tested with 2 to 4 questions per year, spanning fundamental combinatorics, conditional probability, Bayes' theorem, discrete and continuous random variables, and expectation/variance properties. A notable proportion of questions require setting up a discrete probability model (urn/coin/dice experiments or bit strings) or computing expectations using linearity of expectation and integration over piece-wise uniform intervals. The question mix favors NAT and conceptual MCQ/MSQ formats with an emphasis on exact algebraic reasoning rather than heavy numerical statistical testing.

Full Probability and Statistics guide →

More on Probability and Statistics

All 46 questions on Probability and Statistics →

Independent practice explanation verified by the GateAI team. GATE is conducted by the IITs; question text follows the official paper.

Engineering Mathematics · Probability and Statistics

Official IIT answer key · IISc Bangalore · Audited Aug 2026

Take topic test

GATE CSE 2024 Set 1 Q27 · MSQ · 1 mark

Let $A$ and $B$ be two events in a probability space with $P(A) = 0.3$ , $P(B) = 0.5$ , and $P(A \cap B) = 0.1$ . Which of the following statements is/are TRUE?

Select one or more options

Key concept

Probability and Statistics

Step-by-step verified solution

Examiner trap & misconception

Topic notes

How GATE tests Probability and Statistics

Full Probability and Statistics guide →

More on Probability and Statistics

All 46 questions on Probability and Statistics →

Independent practice explanation verified by the GateAI team. GATE is conducted by the IITs; question text follows the official paper.

Let $A$ and $B$ be two events in a probability space with $P(A) = 0.3$ , $P(B) = 0.5$ , and $P(A \cap B) = 0.1$ . Which of the following statements is/are TRUE?

Step-by-step verified solution

Check you can still solve Probability and Statistics under a clock

More from GATE CSE 2024 Set 1

More on Probability and Statistics

Let $A$ and $B$ be two events in a probability space with $P(A) = 0.3$ , $P(B) = 0.5$ , and $P(A \cap B) = 0.1$ . Which of the following statements is/are TRUE?

Step-by-step verified solution

Check you can still solve Probability and Statistics under a clock

More from GATE CSE 2024 Set 1

More on Probability and Statistics

Let AAA and BBB be two events in a probability space with P(A)=0.3P(A) = 0.3P(A)=0.3, P(B)=0.5P(B) = 0.5P(B)=0.5, and P(A∩B)=0.1P(A \cap B) = 0.1P(A∩B)=0.1. Which of the following statements is/are TRUE?

Step-by-step verified solution

Check you can still solve Probability and Statistics under a clock

More from GATE CSE 2024 Set 1

More on Probability and Statistics

Share this solution

Let AAA and BBB be two events in a probability space with P(A)=0.3P(A) = 0.3P(A)=0.3, P(B)=0.5P(B) = 0.5P(B)=0.5, and P(A∩B)=0.1P(A \cap B) = 0.1P(A∩B)=0.1. Which of the following statements is/are TRUE?

Step-by-step verified solution

Check you can still solve Probability and Statistics under a clock

More from GATE CSE 2024 Set 1

More on Probability and Statistics

Share this solution

Let $A$ and $B$ be two events in a probability space with $P(A) = 0.3$ , $P(B) = 0.5$ , and $P(A \cap B) = 0.1$ . Which of the following statements is/are TRUE?

Let $A$ and $B$ be two events in a probability space with $P(A) = 0.3$ , $P(B) = 0.5$ , and $P(A \cap B) = 0.1$ . Which of the following statements is/are TRUE?