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:   (1 -  p₁)p₂  

Answer 1

\[\text { As }, \left( 1 - p_1 \right) p_2 = \left[ 1 - P\left( \overline{ A }\right) \right] \times P\left( B \right) = P\left( A \right) \times P\left( B \right)\]
\[\text{ And, A and B are independent events }. \]
\[\text { i . e} . P\left(\overline{ A } \right) \times P\left( B \right) = P\left( \overline{ A } \cap B \right)\]
\[\text { So }, P\left( \overline{ A } \cap B \right) = \left( 1 - p_1 \right) p_2 \]
\[\text{ Hence} , \left( 1 - p_1 \right) p_2 = P\left( \text{ A does not occur, but B occurs } \right)\]