Statistics And Probability Question 100
Question: If A and B are two events. The probability that at most one of A, B occurs, is
Options:
A) $ 1-P(A\cap B) $
B) $ P(\bar{A})+P(\bar{B})-P(\bar{A}\cap \bar{B}) $
C) $ P(\bar{A})+P(\bar{B})+P(A\cup B)-1 $
D) All of these
Show Answer
Answer:
Correct Answer: D
Solution:
- [d] Required probability $ =P(\bar{A}\cup \bar{B})=P(\overline{A\cap B}) $ $ =1-P(A\cap B) $ Again, $ P(\bar{A}\cup \bar{B})=P(\bar{A})+P(\bar{B})-P(\bar{A}\cap \bar{B}) $ [By add. Theorem] Again, $ P(\bar{A}\cup \bar{B})=P(\bar{A})+P(\bar{B})-P(\bar{A}\cap \bar{B}) $ $ =P(\bar{A})+P(\bar{B})-P(\overline{A\cup B}) $ $ =P(\bar{A})+P(\bar{B})-{1-P(A\cup B)} $ $ =P(\bar{A})+P(\bar{B})+P(A\cup B)-1 $ Finally, $ P(\bar{A}\cup \bar{B})=P[(A\cap \bar{B})\cup (\bar{A}\cap B)\cup (\bar{A}\cap \bar{B})] $ $ =P(A\cap \bar{B})+P(\bar{A}\cap B)+P(\bar{A}\cap \bar{B}) $ [ $ \because A\cap \bar{B},\bar{A}\cap B $ and $ \bar{A}\cap \bar{B} $ are mutually exclusive events] So, alternative [d] is the correct answer.
BETA
