The GATE Grind

GATE 2024 CS (CS1) – Question 27

Engineering Mathematics · Probability and Statistics · 1 mark · Multiple select

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. The two events $A$ and $B$ are independent
  2. $P(A\cup B)=0.7$
  3. $P(A\cap B^c)=0.2$, where $B^c$ is the complement of the event $B$
  4. $P(A^c\cap B^c)=0.4$, where $A^c$ and $B^c$ are the complements of the events $A$ and $B$, respectively

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Show answer and explanation

Correct answer: (B) $P(A\cup B)=0.7$; (C) $P(A\cap B^c)=0.2$, where $B^c$ is the complement of the event $B$

Explanation

P(A)P(B)=0.15≠0.1, so not independent. P(A∪B)=0.3+0.5−0.1=0.7. P(A∩B^c)=0.3−0.1=0.2. P(A^c∩B^c)=1−0.7=0.3, not 0.4.