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Question
If A and B are two independent events such that P (A) = 0.3 and P (A ∪ \[B\]) = 0.8. Find P (B).
Solution
\[\text{ A and B are two independent events. } \]
\[ \therefore P\left( A \cup \bar{B} \right) = P\left( A \right) + P\left( \bar{B} \right) - P\left( A \cap \bar{B} \right)\]
\[ \Rightarrow 0 . 8 = 0 . 3 + \left[ 1 - P\left( B \right) \right] - P\left( A \right) P\left( \bar{B} \right)\]
\[ \Rightarrow 0 . 5 = 1 - P\left( B \right) - 0 . 3\left[ 1 - P\left( B \right) \right]\]
\[ \Rightarrow 0 . 5 = 1 - P\left( B \right) - 0 . 3 + O . 3P\left( B \right)\]
\[ \Rightarrow 0 . 5 = 0 . 7 - P\left( B \right)\left[ 1 - 0 . 3 \right]\]
\[ \Rightarrow 0 . 7P\left( B \right) = 0 . 2\]
\[ \Rightarrow P\left( B \right) = \frac{0 . 2}{0 . 7} = \frac{2}{7}\]
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