$S$ is the sample space and $A, B$ are two events of a random experiment. Match the items of List A with the…

$S$ is the sample space and $A, B$ are two events of a random experiment. Match the items of List A with the items of List B.
  1. (I) - (v), (II) - (iv), (III) - (iii), (IV) - (ii)
  2. (I) - (i), (II) - (iii), (III) - (v), (IV) - (ii)
  3. (I) - (iv), (II) - (iii), (III) - (ii), (IV) - (i)
  4. (I) - (ii), (II) - (iv), (III) - (i), (IV) - (iii)

Solution

I. A, B are mutually exclusive events $\Rightarrow P(A \cap B)=0 \quad \therefore \quad P(A \cup B)=P(A)+P(B)$ II. A, B are independent events $\Rightarrow P(A \cap B)=P(A) \cdot P(B)$ $P\left(\frac{\bar{A}}{B}\right)=\frac{P(\bar{A} \cap B)}{P(B)}=\frac{P(\bar{A}) \cdot P(B)}{P(B)}=P(\bar{A})=1-P(A)$ III. $A \cap B=A \Rightarrow A \subset B \Rightarrow P(A) \leq P(B)$ IV. $A \cup B=S$ $\Rightarrow P(A \cup B)=1 \Rightarrow P(A)+P(B)-P(A \cap B)=1$ $\Rightarrow P(A)-1+P(B)=P(A \cap B)$ $\Rightarrow P(B)-P(\bar{A})=P(A \cap B)$

Asked in: AP EAMCET 2024 (20 May Shift 1)

Practice more Probability questions on Aicharya