JEE MAIN - Mathematics (2002 - No. 27)
$$A$$ and $$B$$ are events such that $$P\left( {A \cup B} \right) = 3/4$$,$$P\left( {A \cap B} \right) = 1/4,$$
$$P\left( {\overline A } \right) = 2/3$$ then $$P\left( {\overline A \cap B} \right)$$ is :
$$P\left( {\overline A } \right) = 2/3$$ then $$P\left( {\overline A \cap B} \right)$$ is :
$$5/12$$
$$3/8$$
$$5/8$$
$$1/4$$
Explanation
Given $$P\left( {A \cup B} \right) = 3/4$$,
$$P\left( {A \cap B} \right) = 1/4,$$
$$P\left( {\overline A } \right) = 2/3$$
We know, $$P\left( A \right)$$ = 1 - $$P\left( {\overline A } \right)$$
$$\therefore$$ $$P\left( A \right)$$ = 1 - $${2 \over 3}$$ = $${1 \over 3}$$
We know $$P\left( {A \cup B} \right)$$ = $$P\left( A \right)$$ + $$P\left( B \right)$$ - $$P\left( {A \cap B} \right)$$
$$ \Rightarrow $$$${3 \over 4}$$ = $${1 \over 3}$$ + $$P\left( B \right)$$ - $${1 \over 3}$$
$$ \Rightarrow $$ 1 = $${1 \over 3}$$ + $$P\left( B \right)$$
$$ \Rightarrow $$ $$P\left( B \right)$$ = $${2 \over 3}$$
We know $$P\left( {\overline A \cap B} \right)$$ = $$P\left( B \right)$$ - $$P\left( {A \cap B} \right)$$
So $$P\left( {\overline A \cap B} \right)$$ = $${2 \over 3}$$ - $${1 \over4}$$ = $${5 \over 12}$$
$$P\left( {A \cap B} \right) = 1/4,$$
$$P\left( {\overline A } \right) = 2/3$$
We know, $$P\left( A \right)$$ = 1 - $$P\left( {\overline A } \right)$$
$$\therefore$$ $$P\left( A \right)$$ = 1 - $${2 \over 3}$$ = $${1 \over 3}$$
We know $$P\left( {A \cup B} \right)$$ = $$P\left( A \right)$$ + $$P\left( B \right)$$ - $$P\left( {A \cap B} \right)$$
$$ \Rightarrow $$$${3 \over 4}$$ = $${1 \over 3}$$ + $$P\left( B \right)$$ - $${1 \over 3}$$
$$ \Rightarrow $$ 1 = $${1 \over 3}$$ + $$P\left( B \right)$$
$$ \Rightarrow $$ $$P\left( B \right)$$ = $${2 \over 3}$$
We know $$P\left( {\overline A \cap B} \right)$$ = $$P\left( B \right)$$ - $$P\left( {A \cap B} \right)$$
So $$P\left( {\overline A \cap B} \right)$$ = $${2 \over 3}$$ - $${1 \over4}$$ = $${5 \over 12}$$
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