Probability of a Misjudged Innocent Suspect Given Truth Serum Results?

[Saladsamurai]

Homework Statement

A truth serum has the property that 90% of the guilty suspects are properly judged while 10% are improperly found innocent. On the other hand innocent suspects are misjudged 1% of the time. If the suspect is selected from a pool of suspects of which only 5% have ever committed a crime, and the serum indicates that he is guilty, what is the probability that he is innocent?

Homework Equations

Not sure yet.

The Attempt at a Solution

So I have denoted the subsets in the sample space as follows:

Let GG denote persons who ARE GUILTY and who are FOUND GUILTY.
Let GI denote persons who ARE GUILTY and who are FOUND INNOCENT.
Let IG denote persons who ARE INNOCENT and who are FOUND GUILTY.
Let A denote the event that an innocent person is picked from the pool.

Am I correct in saying that I am looking for P(IG|A)? That is "what is the probability that an innocent person is misjudged given that an innocent person was selected?"


This is usually the tough part for me, interpreting the question

 

[Dick]

That seems ok to me. If he's going to be innocent then he has to come from the 95% that are innocent and THEN be wrongly found guilty. I'm a little disturbed by the question phrasing. So if you have EVER committed a crime then you are ALWAYS guilty? Doesn't seem quite fair.

 

[Saladsamurai]

That seems ok to me. If he's going to be innocent then he has to come from the 95% that are innocent and THEN be wrongly found guilty. I'm a little disturbed by the question phrasing. So if you have EVER committed a crime then you are ALWAYS guilty? Doesn't seem quite fair.

Yeah, maybe its not the US justice system

What is confusing to me is that this question comes from the section of the text that immediately follows the reading on Bayes' Rule. So with the way this text is set up, I am almost certain that they expect me to use Theorem of total probability (TTP) or BR or some formulation of it.

TTP: If the events B1, B2, ...Bk constitute a partition of the sample space S such that P(Bi) not equal to 0 for all i, then for any event A of S,

$$P(A) = \sum_i P(B_i \cap A) = \sum_i P(B_i)P(A|B_i)$$BR: If the events B1, B2, ...Bk constitute a partition of the sample space S such that P(Bi) not equal to 0 for all i, then for any event A in S such that P(A) <> 0,

$$P(B_r|A) = \frac{P(B_r)P(A|B_r)}{\sum_i P(B_i)P(A|B_i)}$$.

It seems like I am kind of already using one of these... just trying to dot all of my i's here.

EDIT

In words, I can make a tree diagram like so:

Sample Space = {All persons who ARE guilty or INNOCENT}

G = {All GUILTY persons}​

-GG = {All GUILTY persons found Guilty}
-GI = {All Guilty persons found INNOCENT}
​

I = {All INNOCENT persons}​

+II = {All INNOCENT persons found INNOCENT}
+IG = {All INNOCENT persons found GUILTY​

(I just did this to help me visualize any partitioning as I am still trying to figure out how TTP or Bayes' Rule can apply.)

 

[Dick]

I'm not an authority on Bayes, but this doesn't seem to be that deep into Bayesian territory either. I still think your interpretation is correct. If not the ethics of being eternally guilty.

 

[Saladsamurai]

I'm not an authority on Bayes, but this doesn't seem to be that deep into Bayesian territory either. I still think your interpretation is correct. If not the ethics of being eternally guilty.

Okie dokie artichokie!

 

[hgfalling]

Wait, no, I don't agree. This is a very common type of question in basic Bayesian analysis, and I think the question is asking this:

A dude is picked. He is found guilty by the serum. What is the chance that he is innocent?

Your partitioning of the sample space doesn't really have a category for this. Suppose we partition it thusly:

G = all guilty guys
I = all innocent guys
g = guys guilty by the serum
i = guys innocent by the serum

And let Xy be the combinations.

Then you want p(Ig | g); the chance that he is innocent given that he was found guilty by the serum.

 

[Saladsamurai]

Wait, no, I don't agree. This is a very common type of question in basic Bayesian analysis, and I think the question is asking this:

A dude is picked. He is found guilty by the serum. What is the chance that he is innocent?

Your partitioning of the sample space doesn't really have a category for this. Suppose we partition it thusly:

G = all guilty guys
I = all innocent guys
g = guys guilty by the serum
i = guys innocent by the serum

And let Xy be the combinations.

Then you want p(Ig | g); the chance that he is innocent given that he was found guilty by the serum.

So would my Venn diagram be something like:

?

 

[hgfalling]

Well, no, there are guys who are found innocent by the serum who are actually innocent. Actually most of them. In your picture all the guys found innocent by the serum are guilty.

 

[Saladsamurai]

So I would need yet another circle denoted i over in the I subset.

EDIT: Or better yet, more concisely I think that this should cover all combinations:

So now to figure out how to apply Bayes' Rule to this mess. BR is as follows:$$P(B_r|A) = \frac{P(B_r)P(A|B_r)}{\sum_i P(B_i)P(A|B_i)}$$

So I think in this problem B_r is the event that he is innocent and A is the 'given' event that the truth serum indicates he is guilty.

 

[hgfalling]

Yes, just don't forget to recall that the sample space for your problem isn't "all people," it's only the people from the pool of suspects.

 

[Saladsamurai]

Yes, just don't forget to recall that the sample space for your problem isn't "all people," it's only the people from the pool of suspects.

I am not sure what you mean. Do you mean that I need to remember that I am working specifically from a pool in which it is known that 95% are innocent and 5% are guilty?

 

[hgfalling]

That's right. So for the purposes of your problem, anyone outside the 95/5 pool is irrelevant.

 

[Saladsamurai]

So I would need yet another circle denoted i over in the I subset.

EDIT: Or better yet, more concisely I think that this should cover all combinations:

So now to figure out how to apply Bayes' Rule to this mess. BR is as follows:


$$P(B_r|A) = \frac{P(B_r)P(A|B_r)}{\sum_i P(B_i)P(A|B_i)}$$

So I think in this problem B_r is the event that he is innocent and A is the 'given' event that the truth serum indicates he is guilty.

Hmmm...I still do not think that this diagram is appropriate. i think it lacks something, but i am having trouble pinpointing it.

 

[hgfalling]

It doesn't seem to me that Venn diagrams are terribly helpful on this kind of problem.

But yes, Br would be him being innocent and A would be the truth serum indicating his guilt.

You know P(A | Br) from the givens. There are only two "i"s to loop through; he's really guilty or he's really not. The P(A | Bi) and the P(Bi) are in the givens also. So it seems you can plug right into Bayes' rule now?

 

[hgfalling]

Once you do that and get the right answer, I recommend that you try to explain what's going on in your own words also, like you could complete the following Mad Lib (none of the blanks would be percentages unless so noted):

Suppose the sample pool was 1000 suspects. Then ____ of them would be actually guilty, and ____ of them would be innocent. Suppose they were all tested with the serum. Then of the guilty ones, ____ would test guilty, and ____ would test innocent. Of the innocent ones, ___ would test guilty, and ____ would test innocent. So if we tested the entire pool, we'd get ____ guilty tests, and of those guys, ____% would be innocent.

The last blank in the preceding should equal your answer from plugging into Bayes' rule. If it does, you will probably have improved your understanding of this type of problem. :)