# How to calculate the complement of P(T| TT)?

A lie detector will be consulted in a court. It is known that the detector gives the correct result (guilty) for a guilty suspect $90\%$ of the time and it gives the result not guilty for an innocent suspect $99\%$ of the time. From statistics of the tax authorities, it is known that $5\%$ of the citizens cheat seriously in their tax returns . The lie detector indicates that the person is guilty of cheating. What is the probability that the suspect is actually innocent?

$T =$ Tells the truth

$TT =$ Detector thinks it is truth

So I got now, that the probability that someone is telling truth given that the machine thinks is truth, $P(T\mid TT) =$ is around $1881/1891$. But what I need to know, is the probability that someone is actually telling the truth given that the machine thinks it is a lie: $P(T\mid \neg TT)$.

I would simply say that would be $10/1891$, but how can I prove this?

• You may have mixed up your $\neg$s: $P(T \mid \neg TT) = \dfrac{19}{109}$ while $P( \neg T \mid TT) = \dfrac{10}{1891}$. For completeness, $P(\neg T \mid \neg TT) = \dfrac{90}{109}$ and $P( T \mid TT) = \dfrac{1881}{1891}$, the last of which you have calculated – Henry Oct 30 '17 at 14:03

By Bayes' rule we have $$P (T|TT)=\frac {P (TT|T)\cdot P (T)}{P (TT)} .$$ Plugging in all the given numbers we get $P (TT)=\frac {1891}{2000}$ and thus $P ( \neg TT)=\frac {109}{2000}$. By plugging this and the other given numbers into $$P (T|\neg TT)=\frac {P (\neg TT|T)\cdot P (T)}{P (\neg TT)}$$ we get the answer $P (T|\neg TT)=\frac {19}{109}$.