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Question
Let A and B be two independent events such that P(A) = p1 and P(B) = p2. Describe in words the events whose probabilities are: p1 + p2 - 2p1p2
Solution
\[\text{ As } , p_1 + p_2 - 2 p_1 p_2 = \left( p_1 - p_1 p_2 \right) + \left( p_2 - p_1 p_2 \right)\]
\[ = \left[ P\left( A \right) - P\left( A \right) \times P\left( B \right) \right] + \left[ P\left( B \right) - P\left( A \right) \times P\left( B \right) \right]\]
\[\text{ And, A and B are independent events } . \]
\[i . e . P\left( A \right) \times P\left( B \right) = P\left( A \cap B \right)\]
\[ \Rightarrow p_1 + p_2 - 2 p_1 p_2 = \left[ P\left( A \right) - P\left( A \cap B \right) \right] + \left[ P\left( B \right) - P\left( A \cap B \right) \right] = P\left( \text { only } A \right) + P\left( \text { only } B \right)\]
\[\text{ So } , P\left( \text{ only } A \right) + P\left( \text{ only } B \right) = p_1 + p_2 - 2 p_1 p_2 \]
\[\text{ Hence } , p_1 + p_2 - 2 p_1 p_2 = P\left( \text{ Exactly one of A and B occurs } \right)\]
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