Potts has invented a new lie-detector machine and is testing it. One hundred test subjects are told to lie, and the machine catches 80 of them in the lie. Another hundred test subjects are told to tell the truth, but the machine nevertheless thinks that 5 of them are lying.
Let’s say the local police department begins to use the machine in interrogations, and suppose on average 15% of people arrested lie in their interrogations. When the machine indicates a lie, what is the probability that the suspect is really lying? If the machine does not indicate a lie, what is the probability that the suspect is really telling the truth?
My solution
if the probability that a test subject is chosen at random from the 200 in fact lie = 80/(80+5)=0.94
Then when 15% of people arrested lie, the probability that the suspect is really lying is 0.15*0.94=0.141
Is this correct?
Best Answer
No, it is not correct. In fact, I am not sure what that first number you calculate even means ... You say you want to calculate the probability that a person lies ... But we already know that is $0.15$.
Using:
$T$: person teels truth
$TT$ Lie detector thinks person is telling truth
$P(TT) = P(TT|T)*P(T)+P(TT|\neg T)*P(\neg T)= 0.95*0.85+ 0.2*0.15$
Using Bayes's theorem:
$$P(T|TT) = \frac{P(TT|T)*P(T)}{P(TT)} =$$
$$\frac{0.95*0.85}{0.95*0.85+0.2*0.15}$$