If A and B are two events such that P(A B)=(3)/(4) , P(A B)=(1)/(4), P( A )=(2)/(3) where A is the complement of A , then what is P(B) equal to?

Correct answer: B. (2)/(3) P(A B)=(3)/(4), P(A B)=(1)/(4) aligned & P( A )=(2)/(3) P( A )=1-P(A) & P(A)=1-P( A )=1-(2)/(3)=(1)/(3) aligned P(A B)=P(A)+P(B)-P(A B) aligned & (3)/(4)=(1)/(3)+P(B)-(1)/(4) & (3)/(4)+(1)/(4)-(1)/(3)=P(B) aligned aligned & (4)/(4)-(1)/(3)=P(B) & P(B)=1-(1)/(3)=(2)/(3) aligned

NDA 2012

Exam: NDA
Year: 2012
Subject: Mathematics
Chapter: Probability

NDA 2012 · Mathematics · Probability

If A and B are two events such that P(A B)=(3)/(4) , P(A...

If A and B are two events such that

\mathrm{P}(\mathrm{A} \cup \mathrm{B})=\frac{3}{4}

\mathrm{P}(\mathrm{A} \cap \mathrm{B})=\frac{1}{4}, \mathrm{P}(\overline{\mathrm{A}})=\frac{2}{3}

where

\overline{\mathrm{A}}

is the complement of

A

, then what is

\mathrm{P}(\mathrm{B})

equal to?

Options

A $\frac{1}{3}$
B $\frac{2}{3}$
C $\frac{1}{9}$
D $\frac{2}{9}$

Correct answer

B. $\frac{2}{3}$

Step-by-step solution

\mathrm{P}(\mathrm{A} \cup \mathrm{B})=\frac{3}{4}, \mathrm{P}(\mathrm{A} \cap \mathrm{B})=\frac{1}{4}

\begin{aligned} & \mathrm{P}(\overline{\mathrm{A}})=\frac{2}{3} \Rightarrow \mathrm{P}(\overline{\mathrm{A}})=1-\mathrm{P}(\mathrm{A}) \\ & \Rightarrow \mathrm{P}(\mathrm{A})=1-\mathrm{P}(\overline{\mathrm{A}})=1-\frac{2}{3}=\frac{1}{3}\end{aligned}

\mathrm{P}(\mathrm{A} \cup \mathrm{B})=\mathrm{P}(\mathrm{A})+\mathrm{P}(\mathrm{B})-\mathrm{P}(\mathrm{A} \cap \mathrm{B})

\begin{aligned} & \Rightarrow \frac{3}{4}=\frac{1}{3}+\mathrm{P}(\mathrm{B})-\frac{1}{4} \\ & \Rightarrow \frac{3}{4}+\frac{1}{4}-\frac{1}{3}=\mathrm{P}(\mathrm{B})\end{aligned}

$\begin{aligned} & \Rightarrow \frac{4}{4}-\frac{1}{3}=\mathrm{P}(\mathrm{B}) \\ & \Rightarrow \mathrm{P}(\mathrm{B})=1-\frac{1}{3}=\frac{2}{3}\end{aligned}

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