Question

Mark the correct alternative in the following question:

\[\text{ If A and B are two events such that} P\left( A \right) \neq 0 \text{ and }  P\left( B \right) \neq 1,\text{ then } P\left( \overline{ A }|\overline{ B }\right) = \]

Options

• \[ 1 - P\left( A|B \right)\]

• \[ 1 - P\left( \overline{ A }|B \right)\]

• \[\frac{1 - P\left( A \cup B \right)}{P\left( B \right)}\]

• \[\frac{P\left( A \right)}{P\left( B \right)}\]

Answer 1

\[\text{ We have } , \]
\[P\left( A \right) \neq 0 \text{ and }  P\left( B \right) \neq 1\]
\[\text{ Now } , \]
\[P\left( \overline{A }|\overline{B }\right) = \frac{P\left(\overline{ A } \cap\overline{ B } \right)}{P\left( B \right)}\]
\[ = \frac{P\left( \overline{A \cup B }\right)}{P\left( \overline{B} \right)}\]
\[ = \frac{1 - P\left( A \cup B \right)}{P\left( \overline{ B }\right)}\]