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