QJEEM16Madd83: JEE Main 2016 Mathematics - Class 12 JEEM16Madd83 Solution: If A and B are any two events such that P(A) = and P (A B) = ,…

Question

Verified Solution • 2025–26 Session moderate

If A and B are any two events such that P(A) =

{2 \over 5}

and P (A

\cap

B) =

{3 \over {20}}

, hen the conditional probability, P(A

\left| {} \right.

(A'

\cup

B')), where A' denotes the complement of A, is equal to :

Accepted Answer

P\left( {{A \over {A' \cup B'}}} \right)

= {{P\left[ {A \cap \left( {A' \cup B'} \right)} \right]} \over {P\left( {A' \cap B'} \right)}}

= {{P\left[ {\left( {A \cap A'} \right) \cup \left( {A \cap B'} \right)} \right]} \over {P\left( {A \cap B} \right)'}}

\left[ \, \right.

As

\left. {\left( {A \cap B} \right)' = A' \cap B'} \right]

= {{P\left( {A \cap B'} \right)} \over {P\left( {A \cap B} \right)'}}

As

A \cap A' = \phi

\therefore

\phi \cup \left( {A \cap B'} \right) = A \cap B'

= {{P\left( A \right) - P\left( {A \cap B} \right)} \over {1 - P\left( {A \cap B} \right)}}

\left[ \, \right.

As

A \cap B' = A - \left( {A \cap B} \right)

and

\left. {\left( {A \cap B} \right)' = 1 - \left( {A \cap B} \right)} \right]

Given

P\left( A \right) = {2 \over 5}

and

P\left( {A \cap B} \right) = {3 \over {20}}

= {{{2 \over 5} - {3 \over {20}}} \over {1 - {3 \over {20}}}}

= {{{{8 - 3} \over {20}}} \over {{{17} \over {20}}}}

= {5 \over {17}}

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