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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?

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  • $\begingroup$ 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 $\endgroup$ – Henry Oct 30 '17 at 14:03
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By Bayes rule (and the law of total probability for the denominator) \begin{align}P(T\mid \neg TT)&=\frac{P(\neg TT\mid T)P(T)}{P(\neg TT)}\\[0.4cm]&=\frac{P(\neg TT\mid T)P(T)}{P(\neg TT\mid T)P(T)+P(\neg TT\mid \neg T)P(\neg T)}\\[0.4cm]&=\frac{\frac{1}{100}\cdot0.95}{\frac{1}{100}\cdot 0.95+\frac{90}{100}\cdot0.05}\approx 0.174\end{align}

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    $\begingroup$ This is exactly what I just noted down! I feel damn proud, thanks Jimmy $\endgroup$ – Luis E. Oct 30 '17 at 10:57
  • $\begingroup$ Good work @LuisE! $\endgroup$ – Jimmy R. Oct 30 '17 at 10:59
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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} $.

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