Question

If $\overline{E}$ and $\overline{F}$ are the complementary events of events $E$ and $F$, respectively, and if $0<P\left(F\right)<1$, then

• A. $P\left(E/F\right)+P\left(\overline{E}/F\right)=1/2$
• 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: (D)

$P\left(E/\overline{F}\right)+P\left(\overline{E}/\overline{F}\right)=1$

a) $P\left(\frac{E}{F}\right)+P\left(\frac{\overline{E}}{F}\right)=\frac{P(E\bigcap F)}{P\left(F\right)}+\frac{P(\overline{E}\bigcap F)}{P\left(F\right)}$

$=\frac{P(E\bigcap F)+P(\overline{E}\bigcap F)}{P\left(F\right)}=\frac{P\left(F\right)}{P\left(F\right)}=1$

Therefore a) is not correct

b) $P\left(\frac{E}{F}\right)+P\left(\frac{E}{\overline{F}}\right)=\frac{P(E\bigcap F)}{P\left(F\right)}+\frac{P(E\bigcap \overline{F})}{P\left(F\right)}$

$=\frac{P(E\bigcap F)}{P\left(F\right)}+\frac{P(E\bigcap \overline{F})}{1-P\left(F\right)}\ne 1$

Therefore b) is not correct

c) $P\left(\frac{\overline{E}}{F}\right)+P\left(\frac{E}{\overline{F}}\right)=\frac{P(\overline{E}\bigcap F)}{P\left(F\right)}+\frac{P(E\bigcap \overline{F})}{P\left(F\right)}$

$\frac{P(\overline{E}\bigcap F)}{P\left(F\right)}+\frac{P(E\bigcap \overline{F})}{1-P\left(F\right)}\ne 1$

Therefore c) is not correct

d) $P\left(\frac{E}{\overline{F}}\right)+P\left(\frac{\overline{E}}{\overline{F}}\right)=\frac{P(E\bigcap \overline{F})}{P\left(\overline{F}\right)}+\frac{P(\overline{E}\bigcap \overline{F})}{P\left(\overline{F}\right)}$

$=\frac{P(E\bigcap \overline{F})+P(\overline{E}\bigcap \overline{F})}{P\left(\overline{F}\right)}$

$=\frac{P\left(\overline{F}\right)}{P\left(\overline{F}\right)}=1$

Therefore d) is correct.