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 :
Solutions · 1
Official solution
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 \[P\left( {A \cup B} \right)\] = \[P\left( A \right)\] + \[P\left( B \right)\] - \[P\left( {A \cap B} \right)\]
\[ \Rightarrow \] 43=31+P\left( B \right) ( - \]{1 \over 3}⇒1 =31+P\left( B \right) (
\] \Rightarrow P\left( B \right) ( = \]{2 \over 3}We knowP\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}-41=125