Probability Question 205
Question: If A and B are two independent events, then A and $ \bar{B} $ are
Options:
A) Not independent
B) Also independent
C) Mutually exclusive
D) None of these
Show Answer
Answer:
Correct Answer: B
Solution:
Since $ A\cap \bar{B} $ and $ A\cap B $ are mutually exclusive events such that $ A=(A\cap \bar{B})\cup (A\cap B) $
$ P(A)=P(A\cap \bar{B})+P(A\cap B) $
$ \Rightarrow P(A\cap \bar{B})=P(A)-P(A\cap B) $
$ =P(A)-P(A)P(B) $
$ (\because A,B $ are independent)
$ \Rightarrow P(A\cap \bar{B}) $
$ =P(A)(1-P(B))=P(A)P(\bar{B}) $
$ \therefore A $ and $ \bar{B} $ are also independent.
BETA
