Let EC denote the complement of an event E.

Question

Let EC denote the complement of an event E.
Let E1
, E2
 and E3
 be any pairwise independent
events with P(E1) > 0

and P(E1 $$ \cap $$ E2 $$ \cap $$ E3) = 0.

Then P($$E_2^C \cap E_3^C/{E_1}$$) is equal to :

Accepted Answer

Given E1 , E2 , E3 are pairwise indepedent events

so P(E1 $$ \cap $$ E2 ) = P(E1 ).P(E2 )

and P(E2 $$ \cap $$  E3 ) = P(E2 ).P(E3 )

and P(E3 $$ \cap $$ E1 ) = P(E3 ).P(E1 )

and P(E1 $$ \cap $$ E2 $$ \cap $$ E3) = 0

Now $$P\left( {{{E_2^C \cap E_3^C} \over {{E_1}}}} \right)$$

$$ = {{P\left[ {{E_1} \cap \left( {E_2^C \cap E_3^C} \right)} \right]} \over {P\left( {{E_1}} \right)}}$$

$$ = {{P\left( {{E_1}} \right) - \left[ {P\left( {{E_1} \cap {E_2}} \right) + P\left( {{E_1} \cap {E_3}} \right) - P\left( {{E_1} \cap {E_2} \cap {E_3}} \right)} \right]} \over {P\left( {{E_1}} \right)}}$$

$$ = {{P\left( {{E_1}} \right) - P\left( {{E_1}} \right).P\left( {{E_2}} \right) - P\left( {{E_1}} \right).P\left( {{E_3}} \right) - 0} \over {P\left( {{E_1}} \right)}}$$

$$ = 1 - P\left( {{E_2}} \right) - P\left( {{E_3}} \right)$$

$$ = 1 - P\left( {{E_3}} \right) - P\left( {{E_2}} \right)$$

$$ = P\left( {E_3^C} \right) - P\left( {{E_2}} \right)$$

Solutions · 1