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Question
If A and B are independent events, then write expression for P(exactly one of A, B occurs).
Solution
\[ \text{ As, A and B are independent events .} \]
\[\text{ So,} P\left( A \cap B \right) = P\left( A \right) \times P\left( B \right) . . . . . \left( i \right)\]
\[\text{ Now}, \]
\[P\left( \text{ exactly one of A, B occurs } \right) = P\left( \text{ only } A \right) + P\left( \text{ only } B \right)\]
\[ = \left[ P\left( A \right) - P\left( A \cap B \right) \right] + \left[ P\left( B \right) - P\left( A \cap B \right) \right]\]
\[ = \left[ P\left( A \right) - P\left( A \right) \times P\left( B \right) \right] + \left[ P\left( B \right) - P\left( A \right) \times P\left( B \right) \right] \left[ \text{ Using } \left( i \right) \right]\]
\[ = P\left( A \right) \times \left[ 1 - P\left( B \right) \right] + P\left( B \right) \times \left[ 1 - P\left( A \right) \right]\]
\[ = P\left( A \right) \times P\left( \bar{B} \right) + P\left( B \right) \times P\left( \bar{A} \right)\]
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