There are three independent events $A, B$ and $C$ such that probability of occurrence of each event is $p$ . The probability that at least two of the events occur is

$(a) 2 p^{2} - 3 p^{3} (b) 3 p^{2} - 2 p^{3}$
$(c) 3 p^{2} + 2 p^{3} (d) 2 p^{2} + 3 p^{3}$

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Solution

$P (A) = P (B) = P (C) = p$
$P (A^{'}) = P (B^{'}) = P (C^{'}) = 1 - p$

Probability that at least two of the events occur is given by,
$P = P (A \cap B \cap C^{'}) + P (A \cap B^{'} \cap C) + P (A^{'} \cap B \cap C) + P (A \cap B \cap C)$
$= p \times p \times (1 - p) + p \times (1 - p) \times p + (1 - p) \times p \times p + p \times p \times p$
$= 3 p^{2} - 2 p^{3}$
Ans : $(b)$

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