GATE CS 2024 Set 1 — Question 27

Given:
$P(A) = 0.3$
$P(B) = 0.5$
$P(A \cap B) = 0.1$Option (A): Check for independence.
Two events are independent if $P(A \cap B) = P(A) \cdot P(B)$.
$P(A) \cdot P(B) = 0.3 \times 0.5 = 0.15$
Since $P(A \cap B) = 0.1 \neq 0.15$, events $A$ and $B$ are not independent.
So, option (A) is FALSE.Option (B): Calculate $P(A \cup B)$.
Using the formula $P(A \cup B) = P(A) + P(B) - P(A \cap B)$:
$P(A \cup B) = 0.3 + 0.5 - 0.1 = 0.7$
So, option (B) is TRUE.Option (C): Calculate $P(A \cap B^c)$.
$P(A \cap B^c)$ represents the probability of $A$ occurring but not $B$.
$P(A \cap B^c) = P(A) - P(A \cap B) = 0.3 - 0.1 = 0.2$
So, option (C) is TRUE.Option (D): Calculate $P(A^c \cap B^c)$.
By De Morgan's Law, $A^c \cap B^c = (A \cup B)^c$.
$P(A^c \cap B^c) = 1 - P(A \cup B) = 1 - 0.7 = 0.3$
The option states $0.4$, which is incorrect.
So, option (D) is FALSE.