Question

If $\overline{E}$ and $\overline{F}$ are the complementary events of the events $E$ and $F$ respectively then

• A. $P\left(E/F\right)+P\left(\overline{E}/F\right)=1$
• B. $P\left(E/F\right)+P\left(E/\overline{F}\right)=1$
• C. $P\left(\overline{E}/F\right)+P\left(E/\overline{F}\right)=1$
• D. $P\left(E/\overline{F}\right)+P\left(\overline{E}/\overline{F}\right)=1$

Answer 1

Answer: (A), (D)

The correct options are
A $P\left(E/F\right)+P\left(\overline{E}/F\right)=1$
D $P\left(E/\overline{F}\right)+P\left(\overline{E}/\overline{F}\right)=1$

$P\left(E/F\right)+P\left(\overline{E}/F\right)=\frac{P(E\cap F)}{P\left(F\right)}+\frac{P(\overline{E}\cap F)}{P\left(F\right)}=\frac{P(E\cap F)+\{P\left(F\right)-P(E\cap F)\}}{P\left(F\right)}$
since $P(\overline{E}\cap F)=P\left(F\right)-P(E\cap F)$
$\Rightarrow P\left(E/F\right)+P\left(\overline{E}/F\right)=\frac{P\left(F\right)}{P\left(F\right)}=1AlsoP\left(E/\overline{F}\right)+P\left(\overline{E}/\overline{F}\right)=\frac{P(E\cap \overline{F})}{P\left(\overline{F}\right)}+\frac{P(\overline{E}\cap \overline{F})}{P\left(\overline{F}\right)}=\frac{P\left(E\right)-P(E\cap F)+1-P(E\cup F)}{P\left(\overline{F}\right)}=\frac{P\left(E\right)-P(E\cap F)+1-\{P\left(E\right)+P\left(F\right)-P(E\cap F)\}}{P\left(\overline{F}\right)}=\frac{1-P\left(F\right)}{P\left(\overline{F}\right)}=\frac{P\left(\overline{F}\right)}{P\left(\overline{F}\right)}=1$