Let E^C 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 :

Given E₁ , E₂ , E₃ are pairwise indepedent events <br><br>so P(E₁ $\cap$ E₂ ) = P(E₁ ).P(E₂ ) <br><br>and P(E₂ $\cap$ E₃ ) = P(E₂ ).P(E₃ ) <br><br>and P(E₃ $\cap$ E₁ ) = P(E₃ ).P(E₁ ) <br><br>and P(E₁ $\cap$ E₂ $\cap$ E₃) = 0 <br><br>Now $P\left( {{{E_2^C \cap E_3^C} \over {{E_1}}}} \right)$<br><br> $$ = {{P\left[ {{E_1} \cap \left( {E_2^C \cap E_3^C} \right)} \right]} \over {P\left( {{E_1}} \right)}}$$<br><br> $$ = {{P\left( {{E_1}} \right) - \left[ {P\left( {{E_1} \cap {E_2}} \right) + P\left( {{E_1} \cap {E_3}} \right) - P\left( {{E_1} \cap {E_2} \cap {E_3}} \right)} \right]} \over {P\left( {{E_1}} \right)}}$$<br><br> $$ = {{P\left( {{E_1}} \right) - P\left( {{E_1}} \right).P\left( {{E_2}} \right) - P\left( {{E_1}} \right).P\left( {{E_3}} \right) - 0} \over {P\left( {{E_1}} \right)}}$$<br><br> $= 1 - P\left( {{E_2}} \right) - P\left( {{E_3}} \right)$<br><br> $= 1 - P\left( {{E_3}} \right) - P\left( {{E_2}} \right)$<br><br> $= P\left( {E_3^C} \right) - P\left( {{E_2}} \right)$