# The Prosecutor's fallacy and probability

1. Oct 26, 2007

### 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 probabilty?
thx :)

2. Oct 27, 2007

### Hurkyl

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

3. Oct 27, 2007

### sensitive

well the conditional probabilty 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.

4. Oct 27, 2007

### Hurkyl

Staff Emeritus
No prior probabilities were given for what?

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.

5. Oct 27, 2007

### sensitive

Using the conditional probabilty formula,

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

The prior probabilty 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.

6. Oct 27, 2007

### Hurkyl

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

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)

7. Oct 27, 2007

### 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

8. Oct 27, 2007

### Hurkyl

Staff Emeritus
I don't understand where this comes from.

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)

Last edited: Oct 27, 2007
9. Oct 27, 2007

### 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 probabilty that a person is innocent.

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

10. Oct 27, 2007

### Hurkyl

Staff Emeritus
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)

Last edited: Oct 27, 2007
11. Oct 27, 2007

### sensitive

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

sori abt that..

12. Oct 27, 2007

### sensitive

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

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

13. Oct 27, 2007

### Hurkyl

Staff Emeritus
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).

14. Oct 27, 2007

### 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.

15. Oct 27, 2007

### sensitive

whoops let my jus correct a typo;

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

16. Oct 27, 2007

### D H

Staff Emeritus
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

$$P(I|E) = \frac{P(E|I)*P(I)}{P(E|I)*P(I) + P(E|\sim I)*P(\sim I)}$$

Last edited: Oct 27, 2007
17. Oct 27, 2007

### 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

Last edited: Oct 27, 2007
18. Oct 28, 2007

### sensitive

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

Thx for any inputs..:)