Question

If A and B are two events, then P (`overline A` ∩ B) =

Options

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

• 1 − P (A) − P (B)

• P (A) + P (B) − P (A ∩ B)

• P (B) − P (A ∩ B) 

Answer 1

P (B) − P (A ∩ B)

\[\text{ From the diagram, we get A }  \cap B \text{ and}  \bar{A} \cap B \text{ are mutually exclusive events such that } (A \cap B) \cup ( \bar{A} \cap B) = B . \text{ Therefore by addition theorem of probability we have }  \]
\[P(A \cap B) + P( \bar{A} \cap B) = P(B)\]
\[ \therefore P\left( A \cap B \right) = P\left( B \right) - P\left( A \cap B \right)\]