Probability Question 179
Question: A and B are two independent witnesses (i.e there in no collision between them) in a case. The probability that A will speak the truth is x and the probability that B will speak the truth is y. A and B agree in a certain statement. The probability that the statement is true is
Options:
A) $ \frac{x-y}{x+y} $
B) $ \frac{xy}{1+x+y+xy} $
C) $ \frac{x-y}{1-x-y+2xy} $
D) $ \frac{xy}{1-x-y+2xy} $
Show Answer
Answer:
Correct Answer: D
Solution:
A and B will agree in a certain statement if both speak truth or both tell a lie. We define following events $ E_1= $ A and B both speak truth
$ \Rightarrow P(E_1)=xy $
$ E_2= $ A and B both tell a lie
$ \Rightarrow P(E_2)=(1-x)(1-y) $
$ E= $ A and B agree in a certain statement
Clearly, $ P(E/E_1)=1 $ and $ P(E/E_2)=1 $ The required probability is $ P(E_1/E). $
Using Baye’s theorem $ P(E_1/E)=\frac{P(E_1)P(E/E_1)}{P(E_1)P(E/E_1)+P(E_2)P(E/E_2)} $
$ =\frac{xy.1}{xy.1+(1-x)(1-x).1}=\frac{xy}{1-x-y+2xy} $
BETA
