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Question
If A, B and C are independent events such that P(A) = P(B) = P(C) = p, then find the probability of occurrence of at least two of A, B and C.
Solution
\[P\left( \text{ At least two of A, B and C occur } \right) = P\left(\text{ Exactly two of A, B and C occurs } \right) + P\left( \text{ All three occurs } \right)\]
\[ = \left[ P\left( A \cap B \right) - P\left( A \cap B \cap C \right) \right] + \left[ P\left( B \cap C \right) - P\left( A \cap B \cap C \right) \right] + \left[ P\left( A \cap C \right) - P\left( A \cap B \cap C \right) \right] + P\left( A \cap B \cap C \right)\]
\[ = P\left( A \cap B \right) + P\left( B \cap C \right) + P\left( A \cap C \right) - 3P\left( A \cap B \cap C \right) + P\left( A \cap B \cap C \right)\]
\[ = P\left( A \cap B \right) + P\left( B \cap C \right) + P\left( A \cap C \right) - 2P\left( A \cap B \cap C \right)\]
\[ = P\left( A \right) \times P\left( B \right) + P\left( B \right) \times P\left( C \right) + P\left( A \right) \times P\left( C \right) - 2P\left( A \right) \times P\left( B \right) \times P\left( C \right) \left( \text{ As, A, B and C are independent events } \right)\]
\[ = p \times p + p \times p + p \times p - 2p \times p \times p\]
\[ = p^2 + p^2 + p^2 - 2 p^3 \]
\[ = 3 p^2 - 2 p^3\]
So, the probability of occurrence of at least two of A, B and C
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