The Prosecutor's fallacy and probability

[sensitive]

The Prosecutor's fallacy. The probability that there is a DNA match
given that a person is innocent is estimated as 1/100,000. Assume that
the probability that there is a match given that a person is guilty is
1. Suppose that the defendant in a trial lives in a city where there
are 10,000 people that could have committed the crime, and that there
is a DNA match to the defendant. Calculate the probability that the
defendant is indeed guilty, given no other evidence except the DNA
match, i.e., P(guiltyjDNA match). How does this vary as the size of
the population varies?

From the information given above
P(DNA match/guilty) = 1
p(DNA match/innocent) = 1/100000


P(guilty/DNA match) = p(DNA match/guilty)*p(DNA match /innocent)

Am i right? and the no of people; 100000, does that has to do with the probability?
thx :)

 

[Hurkyl]

Whether or not it's right, it's certainly not obvious enough to write without justification! What work led you to that formula?

 

[sensitive]

well the conditional probability formula is p(x/y) = (p(y/x) p(y))/p(x) but in the above scenario no prior probabilities given so i would think that the DNA match for both guilty and innocent are independent hence a product of the two conditional probability.

 

[Hurkyl]

in the above scenario no prior probabilities given

No prior probabilities were given for what?

so i would think that the DNA match for both guilty and innocent are independent hence a product of the two conditional probability.

I don't see how you deduce

P(DNA match) / P(guilty) = P(DNA match | innocent)

nor why you call that a product of two conditional probabilities.

 

[sensitive]

Using the conditional probability formula,

p(Guilty|DNA match) = (p(DNA match|Guilty) P(Guilty))/p(DNA match)


The prior probability I meant was p(Guilty) which is not given

and I am confused with the p(DNAmatch|innocent). I dono wat to do with it or relate it to the conditional formula.

 

[Hurkyl]

Using the conditional probability formula,

p(Guilty|DNA match) = (p(DNA match|Guilty) P(Guilty))/p(DNA match)


The prior probability I meant was p(Guilty) which is not given

But there is other information in the problem you haven't used; maybe it's related?

and I am confused with the p(DNAmatch|innocent).

I dono wat to do with it or relate it to the conditional formula.

There are at least two good options:

(1) Don't do anything until it appears obvious that it can be used. (If it even needs to be used!)

(2) See what other probabilities you can compute from that. (Hoping you'll see an obvious application of those other probabilities)

 

[sensitive]

I am trying my best to get my head around this ques. This is what i came up with.

p(DNA match) = p(DNA match|Guilty)*p(DNA match|innocent)

p(Guilty|DNA match) = (p(DNA match|Guilty) /10000)/(p(DNA match)

I hope I am making sense somewhere.. thx

 

[Hurkyl]

p(DNA match) = p(DNA match|Guilty)*p(DNA match|innocent)

I don't understand where this comes from.

p(Guilty|DNA match) = (p(DNA match|Guilty) /10000)/(p(DNA match)

Don't skip steps! I assume what you forgot to say is that you have decided

P(Guilty) = 1 / 10000?​

I agree; when the problem said you had no additional information, they meant that your a priori on guilt should be uniformly distributed. (i.e. each of the 10,000 are equally likely)

 

[sensitive]

well my understanding is that there are two possibilities that there is a DNA match if a person is guilty and there is also a probability that a person is innocent.

The statement probability that aperson is innocent has nothing to do with the p(guilty) but associated to the p(DNA match)

 

[Hurkyl]

The statement probability that aperson is innocent has nothing to do with the p(guilty)

That's certainly false! A person is innocent if and only if he is not guilty! They are perfectly anti-corrolated!

Nor would that have justified the equation you stated. The equation it would have justified is
P(Guilty and Innocent) = P(Guilty) * P(Innocen)

 

[sensitive]

I overlooked your reply. Yes I meant to say p(guilty) = 1/10000...

sori abt that..

 

[sensitive]

Yes i agree that a person is innocent iff he is not guilty. but there is still a probability of a match although a person is innocent.

therefore shouldn't that be considered in the p(DNA match). Correct me if I am wrong. Thx..

 

[Hurkyl]

Yes, that will be useful. But you won't get the right answer if you translate that fact wrongly into equations! "innocent" is the negation of "guilty"; what formula expresses that fact?

If it helps you think... what is the probability that someone is either innocent or guilty? What about the probability that there is a DNA match and the person is either innocent or guilty?



By the way, it often helps me a lot to eliminate all conditional probabilities when I work these kinds of problems. I usually replace them with their definition:

P(X | Y) = P(X and Y) / P(Y).

 

[sensitive]

probability that someone is either innocent or guilty can be written as

we know P(guilty) = 1/10000 so p(innocent) = 1 - (1/100000)

hence p(guilty or innocent) = p(guilty) + p(innocent)

probability that there is a DNA match and the person is either innocent or guilty can be written as

p(guilty) + p(innocent) - p(guilty and innocent)

= p(guilty) + p(innocent) - [p(guilty)*p( innocent)]

This comes from : the event being guilty or innocent is mutually exclusive in this case.

 

[sensitive]

whoops let my jus correct a typo;

p(innocent) = 1 - p(guilty) = 1 - (1/10000)

 

[D H]

The prosecutor's fallacy results from equating P(evidence|innocent) with P(innocent|evidence), or P(E|I)=P(I|E) for short.

Have you been taught about Baye's Law yet? In this case, Baye's Law says

$$P(I|E) =<br /> \frac{P(E|I)*P(I)}{P(E|I)*P(I) + P(E|\sim I)*P(\sim I)}$$

 

[sensitive]

Regarding the previous question, am I on the right track? Thx.

Whoops sori i didnt know there was a next page.

Well briefly we have been taught on that Bayes theorem. But I should say the formula you gave is not familiar to me or I might have come across under maximum liklihood. I will check again and will have a go with the exercise again. Thank you

 

[sensitive]

Thx you very much..I got the ans. The formula really helps..

Thx for any inputs..:)