If A and B are two events such that P(A)=0.7 , P(B)=0.4 and P(A B )=0.5 , where B denotes the complement of B , then P(B (A B )) is equal:-

Correct answer: A. (1)/(4) aligned & P(A)=(7)/(10), P(B)=(4)/(10) & P(A B )=(5)/(10) & P( B A B )= P(B (A B )) P(A B ) & = P((B B ) (B A)) P(A B ) = P(A B) P(A B ) aligned aligned & = P(A)-P(A B ) P(A)+P( B )-P(A B ) =((7)/(10)-(5)/(10))/((7)/(10)+(1-(4)/(10))-(5)/(10)) & =(2)/(8)=(1)/(4) aligned

JEE Main 20258 AprilEvening Shift

Exam: JEE Main
Year: 2025
Date: 8 April
Shift: Evening Shift
Subject: Mathematics
Chapter: Probability

JEE Main 2025 · 8 April · Evening Shift · Mathematics · Probability

If A and B are two events such that P(A)=0.7 , P(B)=0.4 and P(A B...

A

and

B

are two events such that

P(A)=0.7

\mathrm{P}(\mathrm{B})=0.4

and

\mathrm{P}(\mathrm{A} \cap \overline{\mathrm{B}})=0.5

, where

\overline{\mathrm{B}}

denotes the complement of

B

, then

P(B \mid(A \cup \bar{B}))

is equal:-

Options

A $\frac{1}{4}$
B $\frac{1}{2}$
C $\frac{1}{6}$
D $\frac{1}{3}$

Correct answer

A. $\frac{1}{4}$

Step-by-step solution

\begin{aligned} & \mathrm{P}(\mathrm{A})=\frac{7}{10}, \mathrm{P}(\mathrm{B})=\frac{4}{10} \\ & \mathrm{P}(\mathrm{A} \cup \overline{\mathrm{B}})=\frac{5}{10} \\ & \mathrm{P}\left(\frac{\mathrm{B}}{\mathrm{A} \cup \overline{\mathrm{B}}}\right)=\frac{\mathrm{P}(\mathrm{B} \cap(\mathrm{A} \cup \overline{\mathrm{B}}))}{\mathrm{P}(\mathrm{A} \cup \overline{\mathrm{B}})} \\ & =\frac{\mathrm{P}((\mathrm{B} \cap \overline{\mathrm{B}}) \cup(\mathrm{B} \cap \mathrm{A}))}{\mathrm{P}(\mathrm{A} \cup \overline{\mathrm{B}})}=\frac{\mathrm{P}(\mathrm{A} \cap \mathrm{B})}{\mathrm{P}(\mathrm{A} \cup \overline{\mathrm{B}})}\end{aligned}

\begin{aligned} & =\frac{\mathrm{P}(\mathrm{A})-\mathrm{P}(\mathrm{A} \cap \overline{\mathrm{B}})}{\mathrm{P}(\mathrm{A})+\mathrm{P}(\overline{\mathrm{B}})-\mathrm{P}(\mathrm{A} \cap \overline{\mathrm{B}})}=\frac{\frac{7}{10}-\frac{5}{10}}{\frac{7}{10}+\left(1-\frac{4}{10}\right)-\frac{5}{10}} \\ & =\frac{2}{8}=\frac{1}{4}\end{aligned}

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