Math Help - Conditional Probability and Bayes' Theorem

Hello I was hoping someone could give me a hand with this question.

A crime is committed, and DNA evidence is discovered. The DNA is compared with the national database and one match is found. In court, the prosecutor tells the jury that the probability of seeing this match if the suspect is innocent is 1 in 1 000 000. Assume that only very vague extra information is known about the suspect, so, except for the DNA data, there is a pool of 5 000 000 equally likely suspects.

Denote by D = {match in the DNA database} and by G = {the suspect is guilty}.

(a) Write down P(G), P(D|G) and P(D|G') [G' = complement of G]
(b) Compute the probability that the suspect is guilty.

As far as I can see P(D|G') = 1/1 000 000 ... but I can't work out the other probabilities as the 5 million equally likely suspects is confusing me. Also I think in part (b) it means P(G|D)... any help would be great and appreciated.

Originally Posted by slevvio

Hello I was hoping someone could give me a hand with this question.

A crime is committed, and DNA evidence is discovered. The DNA is compared with the national database and one match is found. In court, the prosecutor tells the jury that the probability of seeing this match if the suspect is innocent is 1 in 1 000 000. Assume that only very vague extra information is known about the suspect, so, except for the DNA data, there is a pool of 5 000 000 equally likely suspects.

Denote by D = {match in the DNA database} and by G = {the suspect is guilty}.

(a) Write down P(G), P(D|G) and P(D|G') [G' = complement of G]
(b) Compute the probability that the suspect is guilty.

As far as I can see P(D|G') = 1/1 000 000 ... but I can't work out the other probabilities as the 5 million equally likely suspects is confusing me. Also I think in part (b) it means P(G|D)... any help would be great and appreciated.

P(G)=1/5000000 ..the apriori probability of guilt

I think you are supposed to assume that P(D|G) is close to 1

P(D|G')=1/1000000 .. given

Then:

P(G|D)=P(D|G)P(G)/P(D)=P(D|G)P(G)/[P(D|G)P(G)+P(D|G')P(G')]=

.... 1(1/5000000)/[(1/5000000)+(1/1000000)(4999999/5000000)] ~=1/6

(I think, check the arithmetic and logic)

CB