Conditional Probability in a judge/suspect problem

Question

The verdict given by a judge is 90% reliable when a suspect is guilty and 99% reliable when the suspect is innocent. If the suspect is selected from a group of people of whom 5% have ever committed a crime and the judge finds him guilty, what is the probability that the suspect is innocent?

My job: Let $I$ the event "the suspect is innocent". Then

$\begin{eqnarray*} P(I)&=&P(I|\mbox{Judge is reliable})P(\mbox{Judge is reliable})\\ &&+P(I|\mbox{Judge is not reliable})P(\mbox{Judge is not reliable})\\ \end{eqnarray*}$

On the other hand, I know $P(\mbox{Judge is reliable}|I^c)=0.9$ and $P(\mbox{Judge is reliable}|I)=0.99$. Now, I don't know how to use the bold hypothesis. I don't know how to proced.

Answer 1

This is an application of Bayes' theorem!

$P(I) = P(I\mid\text{Guilty})P(\text{Guilty})$

and we know that $P(\text{Guilty})=0.05$.

Now, using Bayes' theorem,

\begin{equation} P(I\mid\text{Guilty}) = P(\text{Guilty}\mid I) \frac{P(I)}{P(\text{Guilty})} \end{equation}

In our population, the probability of innocent is $0.95$ and the probability of guilty is $0.05$; therefore $P(I)/P(\text{Guilty})=0.95/0.05$. And $P(\text{Guilty}\mid I)=0.01$. So now we can use Bayes' theorem to find $P(I\mid\text{Guilty})$, and we can plug that in our first equation to get the answer.

Answer 2

You want the conditional probability that the Suspect is Innocent given the Judge found him Guilty.

The suspect will be found Guilty if either: Is Innocent and Judge not reliable, or Is not Innocent and the Judge is reliable.

$$\mathsf P(I\mid G)=\dfrac{\mathsf P(I\cap G)}{\mathsf P(G)}=\dfrac{\mathsf P(I\cap J^\complement)}{\mathsf P(I\cap J^\complement)+\mathsf P(I^\complement\cap J)}$$