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)

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

Answer 2

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