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Question
Mark the correct alternative in the following question:
\[\text{ If A and B are such that } P\left( A \cup B \right) = \frac{5}{9} \text{ and } P\left( \overline{A} \cup \overline{B} \right) = \frac{2}{3}, \text{ then } P\left( A \right) + P\left( B \right) = \]
Options
\[\frac{9}{10}\]
\[\frac{10}{9}\]
\[ \frac{8}{9} \]
\[\frac{9}{8}\]
Solution
\[ \text{ We have } , \]
\[P\left( A \cup B \right) = \frac{5}{9} \text{ and } P\left( \overline{A} \cup \overline{B} \right) = \frac{2}{3}\]
\[\text{ As } , P\left( \overline{A} \cup \overline{B} \right) = \frac{2}{3}\]
\[ \Rightarrow P\left( \overline{ A \cap B }\right) = \frac{2}{3}\]
\[ \Rightarrow P\left( A \cap B \right) = 1 - \frac{2}{3}\]
\[ \Rightarrow P\left( A \cap B \right) = \frac{1}{3}\]
\[ \text{ Also } , \]
\[P\left( A \cup B \right) = P\left( A \right) + P\left( B \right) - P\left( A \cap B \right)\]
\[ \Rightarrow P\left( A \right) + P\left( B \right) = P\left( A \cup B \right) + P\left( A \cap B \right)\]
\[ = \frac{5}{9} + \frac{1}{3}\]
\[ \Rightarrow P\left( A \right) + P\left( B \right) = \frac{8}{9} . . . . . \left( i \right)\]
\[\text{ Now } , \]
\[P\left( \overline{A} \right) + P\left( \overline{B} \right) = 1 - P\left( A \right) + 1 - P\left( B \right)\]
\[ = 2 - \left[ P\left( A \right) + P\left( B \right) \right]\]
\[ = 2 - \frac{8}{9} \left[ \text{ Using } \left( i \right) \right]\]
\[ = \frac{10}{9}\]
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