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Impossible Events

If A, B and C...

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.

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Solution

$P (At least two of A, B and C occur) = P (Exactly two of A, B and C occurs) + P (All three occurs) = [P (A \cap B) - P (A \cap B \cap C)] + [P (B \cap C) - P (A \cap B \cap C)] + [P (A \cap C) - P (A \cap B \cap C)] + P (A \cap B \cap C) = P (A \cap B) + P (B \cap C) + P (A \cap C) - 3 P (A \cap B \cap C) + P (A \cap B \cap C) = P (A \cap B) + P (B \cap C) + P (A \cap C) - 2 P (A \cap B \cap C) = P (A) \times P (B) + P (B) \times P (C) + P (A) \times P (C) - 2 P (A) \times P (B) \times P (C) (As, A, B and C are independent events) = p \times p + p \times p + p \times p - 2 p \times p \times p = p^{2} + p^{2} + p^{2} - 2 p^{3} = 3 p^{2} - 2 p^{3}$

So, the probability of occurrence of at least two of A, B and C $3 p^{2} - 2 p^{3}$ .

Disclaimer: The answer given in the book is incorrect. the same has been corrected above.

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