Results 1 to 4 of 4

Thread: Probability of the correct diagnosis in a lie detector

  1. #1
    Newbie
    Joined
    Oct 2017
    From
    DR
    Posts
    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?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    Dec 2013
    From
    Colombia
    Posts
    1,840
    Thanks
    592

    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).
    Last edited by Archie; Oct 4th 2017 at 06:39 PM.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor

    Joined
    Aug 2006
    Posts
    21,499
    Thanks
    2730
    Awards
    1

    Re: Probability of the correct diagnosis in a lie detector

    Quote Originally Posted by karlinkas View Post
    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*}$
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor

    Joined
    Apr 2005
    Posts
    19,462
    Thanks
    2893

    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".
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Conditional Probability.. Am I correct?
    Posted in the Statistics Forum
    Replies: 1
    Last Post: Oct 28th 2014, 11:43 PM
  2. Replies: 15
    Last Post: Jul 29th 2011, 02:39 AM
  3. Locating the mirror and detector
    Posted in the Trigonometry Forum
    Replies: 3
    Last Post: May 24th 2010, 10:21 PM
  4. Probability of a Lie Detector
    Posted in the Advanced Statistics Forum
    Replies: 3
    Last Post: Sep 10th 2009, 04:04 AM
  5. Probability (Is what I did correct?)
    Posted in the Advanced Statistics Forum
    Replies: 3
    Last Post: Nov 4th 2007, 11:31 PM

/mathhelpforum @mathhelpforum