# Bayes' theorem - Polygraph

This is a Bayes' theorem question but I'm kind of confused how to tackle it.

Someone lying fails the polygraph 95% of the time. However, someone telling the truth also fails 10% of the times. If a polygraph indicates that the applicant is lying, what is the probability that he is telling the truth?

(Assume a general probability $p$ that the person is truthful)

HINT

Key to these problems is clear organization: Clearly define the relevant events, the probabilities given to you, and the probability you are looking for. For example:

$T$: person is telling the truth

$P$: polygraph indicates person is telling the truth ('passes the polygraph')

With these, you are given:

$P(\neg P|\neg T)=0.95$ (and thus $P(P|\neg T)=1-0.95=0.05$)

$P(\neg P|T)=0.1$ (and thus $P(P|T)=1-0.1=0.9$)

$P(T)=p$ (and thus $P(\neg T) = 1-p$)

and you are looking for:

$P(T|\neg P)$

Now, do you know the Bayesian formula?