Statistics And Probability Question 372
Question: If $ \bar{E} $ and $ \bar{F} $ are the complementary events of events E and F respectively and if $ 0<P,(F)<1, $ then
[IIT 1998]
Options:
A) $ P,(E/F)+P,(\bar{E}/F)=1 $
B) $ P,(E/F)+P,(E/\bar{F})=1 $
C) $ P,(\bar{E}/F)+P,(E/\bar{F})=1 $
D) $ P,(E/\bar{F})+P,(\bar{E}/\bar{F})=1 $
Show Answer
Answer:
Correct Answer: A
Solution:
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$ P(E/F)+P(\bar{E}/F)=\frac{P(E\cap F)+P(\bar{E}\cap F)}{P(F)} $ $ =\frac{P\{(E\cap F)\cup (\bar{E}\cap F)\}}{P(F)} $ $ [\because \ E\cap F $ and $ \bar{E}\cap F $ are disjoint] $ =\frac{P\{(E\cup \bar{E})\cap F\}}{P(F)}=\frac{P(F)}{P(F)}=1 $ Similarly we can show that and are not true while is true. $ P( \frac{E}{{\bar{F}}} )+P( \frac{{\bar{E}}}{{\bar{F}}} )=\frac{P(E\cap \bar{F})}{P(F)}+\frac{P(\bar{E}\cap \bar{F})}{P(F)}=\frac{P(\bar{F})}{P(\bar{F})}=1 $
BETA
