Question
Let $$A$$ and $$B$$ be two events. Then $$1 + P\left( {A \cap B} \right) - P\left( B \right) - P\left( A \right)$$ is equal to :
A.
$$P\left( {\overline A \cup \overline B } \right)$$
B.
$$P\left( {A \cap \overline B } \right)$$
C.
$$P\left( {\overline A \cap B} \right)$$
D.
$$P\left( {\overline A \cap \overline B } \right)$$
Answer :
$$P\left( {\overline A \cap \overline B } \right)$$
Solution :
$$\eqalign{
& {\text{We know,}} \cr
& P\left( {A \cup B} \right) = P\left( A \right) + P\left( B \right) - P\left( {A \cap B} \right) \cr
& {\text{Consider}}\,;\,1 + P\left( {A \cap B} \right) - P\left( B \right) - P\left( A \right) \cr
& = 1 - P\left( B \right) - P\left( A \right) + P\left( A \right) + P\left( B \right) - P\left( {A \cup B} \right) \cr
& = 1 - P\left( {A \cup B} \right) \cr} $$