For two events A and B , if P(A)=P( A B )=(1)/(4) and P((B)/(A))=(1)/(2) , then which of the following is not true?

Correct answer: D. None of these P( B A )=(1)/(2) P(B A) P(A) =(1)/(2) P(B A)=(1)/(8)P( A B )=(1)/(4) P(A B) P(B) =(1)/(4) P(B)=(1)/(2)P(A B)=(1)/(8)=P(A) · P(B) Events A and B are independent. Now, P( A^ B )= P(A B) P(B) = P(A) P(B) P(B) =(3)/(4) and P( B^ A^ )= P(B^ A^ ) P(A^ ) = P (B^ ) P(A^ ) P(A^ ) =(1)/(2)

BITSAT 2024Paper 1

Exam: BITSAT
Year: 2024
Paper: Paper 1
Subject: Mathematics
Chapter: Probability

BITSAT 2024 · Paper 1 · Mathematics · Probability

For two events A and B , if P(A)=P( A B )=(1)/(4) and...

For two events A and B , if

\mathrm{P}(\mathrm{A})=\mathrm{P}\left(\frac{\mathrm{A}}{\mathrm{B}}\right)=\frac{1}{4} and

P\left(\frac{B}{A}\right)=\frac{1}{2}$, then which of the following is not true?

Options

AA and B are independent
B\mathrm{P} $\left(\frac{\mathrm{A}^{\prime}}{\mathrm{B}}\right)=\frac{3}{4}$
C\mathrm{P} $\left(\frac{\mathrm{B}^{\prime}}{\mathrm{A}^{\prime}}\right)=\frac{1}{2}$
DNone of these

Correct answer

D. None of these

Step-by-step solution

\mathrm{P}\left(\frac{\mathrm{B}}{\mathrm{A}}\right)=\frac{1}{2} \Rightarrow \frac{\mathrm{P}(\mathrm{B} \cap \mathrm{A})}{\mathrm{P}(\mathrm{A})}=\frac{1}{2} \Rightarrow \mathrm{P}(\mathrm{B} \cap \mathrm{A})=\frac{1}{8}\mathrm{P}\left(\frac{\mathrm{A}}{\mathrm{B}}\right)=\frac{1}{4} \Rightarrow \frac{\mathrm{P}(\mathrm{A} \cap \mathrm{B})}{\mathrm{P}(\mathrm{B})}=\frac{1}{4} \Rightarrow \mathrm{P}(\mathrm{B})=\frac{1}{2}\mathrm{P}(\mathrm{A} \cap \mathrm{B})=\frac{1}{8}=\mathrm{P}(\mathrm{A}) \cdot \mathrm{P}(\mathrm{B})\therefore

Events A and B are independent. Now,

\mathrm{P}\left(\frac{\mathrm{A}^{\prime}}{\mathrm{B}}\right)=\frac{\mathrm{P}(\mathrm{A} \cap \mathrm{B})}{\mathrm{P}(\mathrm{B})}=\frac{\mathrm{P}(\mathrm{A}) \mathrm{P}(\mathrm{B})}{\mathrm{P}(\mathrm{B})}=\frac{3}{4}

and

\mathrm{P}\left(\frac{\mathrm{B}^{\prime}}{\mathrm{A}^{\prime}}\right)=\frac{\mathrm{P}\left(\mathrm{B}^{\prime} \cap \mathrm{A}^{\prime}\right)}{\mathrm{P}\left(\mathrm{A}^{\prime}\right)}

=\frac{\mathrm{P}

\left(\mathrm{B}^{\prime}\right) \mathrm{P}\left(\mathrm{A}^{\prime}\right)}{\mathrm{P}\left(\mathrm{A}^{\prime}\right)}=\frac{1}{2}

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