Thread: Probability of the correct diagnosis in a lie detector

1. Probability of the correct diagnosis in a lie detector

A lie detector correctly diagnoses 90% of those who lie (M) and 95% of those who do not lie. A person is chosen at random from a group of 100 people from which 20 are known to lie. Whether that person lies or not, what is the probability that the detector will provide a correct diagnosis?

2. Re: Probability of the correct diagnosis in a lie detector

DL / DH = Diagnosed Liar / Honest
AL / AH = Actual Liar / Honest

$$\begin{array}{ r | c | c } & DH & DL \\ \hline AH & 76 & 4 \\ \hline AL & 2 & 18 \end{array}$$
Two of those cells are correct diagnoses.

More interesting is the probability that someone is a liar given a diagnosis of "liar". Especially if the number of liars is very small compared to the population (20 in 1000, say).

3. Re: Probability of the correct diagnosis in a lie detector

A lie detector correctly diagnoses 90% of those who lie (M) and 95% of those who do not lie. A person is chosen at random from a group of 100 people from which 20 are known to lie. Whether that person lies or not, what is the probability that the detector will provide a correct diagnosis?
Notation: $C$ is a correct diagnoses; if $M$ is known liar then $\neg M$ is not a liar.
\begin{align*}\mathcal{P}(C)&=\mathcal{P}(C\cap M)+\mathcal{P}(C\cap\neg M) \\&=\mathcal{P}(C|M)\mathcal{P}(M)+\mathcal{P}(C| \neg M)\mathcal{P}( \neg M)\\&= (?)(?)+(?)(?)\\&=~? \end{align*}

4. Re: Probability of the correct diagnosis in a lie detector

If you had tested all 100 people, 90% of the 20 people who would lie, 18, would test as "liars" and the other 2 would not. 95% of the 80 who do not lie, 76, would test as non-liars, 4 would test a liars. Of those 100 people, 18+ 76= 94, or 94% of the 100 people tested "correctly".