**Question **: If E’ and F’ are the complementary events of events E and F, respectively, and if 0<P(F)<1, then

(A) P(E/F)+P(E’/F)=1/2

(B) P(E/F)+P(E/F’)=1

(C) P(E’/F)+P(E/F’)=1

(D) P(E/F’)+P(E’/F’)=1

**Answer **: option (D)

**Solution **: P\left( E/F\right) +P\left( \overline{E}| F\right) =\dfrac{P\left( E\cap F\right) +P\left( \overline{E}\cap F\right) }{P\left( F\right) }
=\dfrac{P\left( E\cap F\right) +P\left( E^{1}\cap F\right) }{P\left( F\right) }

(because EnF and E’nF are disjoint)

=P{(EUE’)nF}/P(F) = P(F)/P(F) =1

Similarly, we can show that (b) and (c) are not true while (d) is true

\begin{aligned}P\left( E| F^{1}\right) +P\left( E^{1}| F^{1}\right) =\dfrac{P\left( E\cap F^{1}\right) }{P\left( F^{1}\right) }+\dfrac{P\left( E^{1}\cap F^{1}\right) }{P\left( F^{1}\right) }=\ \dfrac{P\left( F^{1}\right) }{P\left( F^{1}\right) }=1\end{aligned}