GATE CS 2024 Set 1 | Question 27

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?

  1. Option A: The two events \(A\) and \(B\) are independent
  2. Option B: \(P(A \cup B) = 0.7\) (correct)
  3. Option C: \(P(A \cap B^{c}) = 0.2\), where \(B^{c}\) is the complement of the event \(B\) (correct)
  4. Option D: \(P(A^{c} \cap B^{c}) = 0.4\), where \(A^{c}\) and \(B^{c}\) are the complements of the events \(A\) and \(B\), respectively

Answer

(B), (C)

Explanation

  • A: false. Independence needs \(P(A \cap B) = P(A)P(B) = 0.3 \times 0.5 = 0.15\), but \(P(A \cap B) = 0.1\).
  • B: true. \(P(A \cup B) = P(A) + P(B) - P(A \cap B) = 0.3 + 0.5 - 0.1 = 0.7\).
  • C: true. \(A\) splits into the disjoint parts \(A \cap B\) and \(A \cap B^{c}\), so \(P(A \cap B^{c}) = 0.3 - 0.1 = 0.2\).
  • D: false. By De Morgan's law \(A^{c} \cap B^{c} = (A \cup B)^{c}\), so \(P(A^{c} \cap B^{c}) = 1 - 0.7 = 0.3\), not 0.4.

Answer: B, C