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Question
If P (not B) = 0.65, P (A ∪ B) = 0.85, and A and B are independent events, then find P (A).
Solution
\[P\left( \bar{B} \right) = 0 . 65\]
\[ \Rightarrow 1 - P\left( B \right) = 0 . 65\]
\[ \Rightarrow P\left( B \right) = 1 - 0 . 65 = 0 . 35\]
\[\text{ Now } , \]
\[P\left( A \cup B \right) = P\left( A \right) + P\left( B \right) - P\left( A \cap B \right)\]
\[ \Rightarrow P\left( A \cup B \right) = P\left( A \right) + P\left( B \right) - P\left( A \right) \times P\left( B \right)\]
\[ \Rightarrow 0 . 85 = P\left( A \right) + 0 . 35 - 0 . 35 \times P\left( A \right)\]
\[ \Rightarrow 0 . 85 - 0 . 35 = P\left( A \right)\left[ 1 - 0 . 35 \right]\]
\[ \Rightarrow P\left( A \right) = \frac{0 . 5}{0 . 65} = 0 . 77\]
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