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 - (1 - p_1 )(1 -p_2 ) `

Answer 1

\[\text {  As, } 1 - \left( 1 - p_1 \right)\left( 1 - p_2 \right) = 1 - \left[ 1 - P\left( A \right) \right] \times \left[ 1 - P\left( B \right) \right]\]
\[ = 1 - P\left( \overline{ A } \right) \times P\left( \overline{ B } \right)\]
\[\text{ And, A and B are independent events }  . \]
\[\text { i . e }. P\left( \overline{ A } \right) \times P\left( \overline{ B } \right) = P\left(\overline{ A } \cap \overline{ B } \right)\]
\[ \Rightarrow 1 - \left( 1 - p_1 \right)\left( 1 - p_2 \right) = 1 - P\left( \overline{ A } \cap\overline{ B } \right) = 1 - P\left( \overline{ A \cup B } \right) = P\left( A \cup B \right)\]
\[\text{ So } , P\left( A \cup B \right) = 1 - \left( 1 - p_1 \right)\left( 1 - p_2 \right)\]
\[ \text{ Hence } , 1 - \left( 1 - p_1 \right)\left( 1 - p_2 \right) = P\left( \text{ At least one of A and B occurs } \right)\]