Let A, B and C be three events, which are pair-wise independent and [ E ] denotes the comp…

Question

Let A, B and C be three events, which are pair-wise independent and $$\overrightarrow E $$ denotes the completement of an event E. If $$P\left( {A \cap B \cap C} \right) = 0$$ and $$P\left( C \right) > 0,$$ then $$P\left[ {\left( {\overline A  \cap \overline B } \right)\left| C \right.} \right]$$ is equal to :

Accepted Answer

Here, $$P\left( {\overline A  \cap \overline B \left| C \right.} \right) = {{P\left( {\overline A  \cap \overline B  \cap C} \right)} \over {P\left( C \right)}}$$

= $${{P\left[ {\left( {\overline {A \cup B} } \right) \cap C} \right]} \over {P\left( C \right)}}$$

= $${{P\left[ {C - \left( {A \cup B} \right)} \right]} \over {P\left( C \right)}}$$

= $${{P\left( C \right) - P\left( {A \cap C} \right) - P\left( {B \cap C} \right) + P\left( {A \cap B \cap C} \right)} \over {P\left( C \right)}}$$

= $${{P\left( C \right) - P\left( {A \cap C} \right) - P\left( {B \cap C} \right)} \over {P\left( C \right)}}$$ ($$ \because $$ $\left. {P\left( {A \cap B \cap C} \right) = 0} \right) (

= \]{{P\left( C \right) - P\left( A \right).P(C) - P\left( B \right).P(C)} \over {P\left( C \right)}}$$

[$$ \because $$ A, B and C are independent events]

= 1 - P(A) - P(B)

= $$P\left( {\overline A } \right) ( - P(B) or \]P\left( {\overline B } \right)$

Solutions · 1