Browse by Stream
QnA
Home
QnA
Go To Your Study Dashboard

Get Answers to all your Questions

header-bg qa
  • Home
  • NCERT
  • State True or False for the statements in the Exercise. If A and B are independent events, then A′ and B′ are also independent.

State True or False for the statements in the Exercise.
If A and B are independent events, then A′ and B′ are also independent.

Answers (1)

TRUE

As A and B are independent

\begin{aligned} &\Rightarrow P({A} \cap {B})=P(A) P(B)\\ &\mathrm{P}\left(\mathrm{A}^{\prime} \cap \mathrm{B}^{\prime}\right)=\mathrm{P}({\mathrm{A}} \underline{\cup} \underline{\mathrm{B}})^{\prime}\{\text { using De morgan's law }\}\\ &P(A \cup B)^{\prime}=1-P(A \cup B)\\ &\text { We know } P(A \cup B)=P(A)+P(B)-P(A \cap B)\\ &\Rightarrow P(A \cup B)^{\prime}=1-[P(A)+P(B)-P(A \cap B)]\\ &\Rightarrow \mathrm{P}\left(\mathrm{A}^{\prime} \cap \mathrm{B}^{\prime}\right)=1-\mathrm{P}(\mathrm{A})-\mathrm{P}(\mathrm{B})+\mathrm{P}(\mathrm{A}) \mathrm{P}(\mathrm{B}) \text { as } \mathrm{A} \& \mathrm{~B} \text { are independent }\}\\ &=[1-\mathrm{P}(\mathrm{A})]-\mathrm{P}(\mathrm{B})(1-\mathrm{P}(\mathrm{A})]\\ &\Rightarrow P\left(A^{\prime} \cap B^{\prime}\right)=(1-P(A))(1-P(B))\\ &=\mathrm{P}\left(\mathrm{A}^{\prime}\right) \mathrm{P}\left(\mathrm{B}^{\prime}\right) \end{aligned}
hence proved

Posted by

infoexpert22

View full answer

Similar Questions

Latest Question