Question

If $E$ and $F$ are events such that $0<P(F)<1$, then

• A. $P\left(E|F\right)+P\left(\bar{E}|F\right)=1$
• B. $P\left(E|F\right)+P\left(E|\bar{F}\right)=1$
• C. $P\left(\bar{E}|F\right)+P\left(E|\bar{F}\right)=1$
• D. $P\left(E|\bar{F}\right)+P\left(\bar{E}|\bar{F}\right)=0$

Answer 1

Answer: (A)

$P\left(E|F\right)+P\left(\bar{E}|F\right)$
$=\frac{P\left(E\cap F\right)+P\left(\bar{E}\cap F\right)}{P\left(F\right)}$
$=\frac{P\left(\left(E\cup \bar{E}\right)\cap F\right)}{P\left(F\right)}$
$=\frac{P\left(F\right)}{P\left(F\right)}=1$