# DNA use in court cases

I am currently studying the following case of Neil Owen, based on the following article I found a newspaper:

"A 20-year-old student was jailed for life yesterday for the brutal rape and murder of a schoolgirl, after one of the biggest DNA testing programmes in British criminal history. Neil Owen was arrested a year after the murder when his genetic fingerprint was matched with DNA found at the scene, following a mass DNA screening of 2000 men on the estate. He lived just 100 yards from the victim's house. Laboratory tests revealed the chances of anyone else being the killer were 1 in 160 million."

Now first of all I am aware that there is an issue with prosecutors fallacy here. Because the 1 in 160 million is interpreted as P(innocence|matching blood type evidence) when it actually refers to P(matching blood type evidence|innocence). But my question refers to the reasoning of the defence.

Counsel for the defence pointed out that there are about 30 million males in the United Kingdom, and argued that the correct probability that Owen was guilty is about 16/19, not high enough to convict beyond a reasonable doubt. So my two questions are

1. How do you think that the figure 16/19 was calculated? (I am sure the population of 30 million and the probability of 1 in 160 million was used?)

2. What implicit assumptions were made, and how reasonable are they?

-
Not that this makes the question any less important or interesting, but from reports I've found on the web, it appears that Owen claimed they had consensual sex and somebody else murdered her - so I suppose that the DNA evidence was not an important factor in his conviction (except that his connection to the case would not have been discovered without it)? –  James May 14 '12 at 13:34

1. Given your assumption that 1 in 160 millions being P(matching DNA evidence|random person), the number 16/19 is roughly the chance that none of the other 30 million males in the UK would also match the DNA evidence: binomial chance of 0 hits, given 30 millions trials with p = 1/160 millions. I get about 0.83 for this probability and 16/19 is roughly 0.84. Since 19/23 is a better approximation of the probability I calculated I am not certain if this is how they got it.
2. Assumptions of whom? The counsel? If I am right, he makes the incorrect assumption that the existence of another man with matching DNA would mean that his client is inoccent. But of the 30 million men many would have alibis and/or life far away from the crime scene which gives them a relative miniscule prior probability of being the killer.

Statistically it makes sense to assume he's guilty. If we had a measure of often the killer lives near and therefore how likely it is that he is part of the 2000 people tested, we could calculate the probability. Let's say it's relatively low, say 5%. Let be G be the event that the guilty one is part of the 2000 and let E be the event that at least one of the 2000 tests positive.

Then

$$P(G|E)=\frac{P(E|G)P(G)}{P(E)}.$$

P(G) is assumed to be 0.05 and $P(E|G)$ should be about 1 if the lab does it's work properly. In practice it's probably slightly lower, so let's assume it's just 0.9. OTH $$P(E) = P(E|G)P(G)+P(E|!G)P(!G)=0.05*0.9+ p*0.95$$ with p being the binomial chance of at least 1 positive result out of 2000 with a hit chance of 1/160 million. It turns out that this is small, with p being about $0.000012$. This means we get $P(E)=0.045$ and $$P(G|E) \approx 0.99974.$$

-
Very interesting. An alternative defence argument is that, since 2000 men were examined, the probability of nding an innocent person who matched the DNA at the scene was 20001=160 million, 1/80,000, greatly weakening the strength of the match evidence. Prosecution counters that the search in fact eliminated 1999 individuals who might otherwise have been considered as possible alternative perpetrators, and so strengthened the evidence against Owen. Are these arguments correct? –  Valerie May 14 '12 at 12:01
It's roughly 1/800000. You are off by a factor of 10. You can find that number above as p. I don't think a small number like that weakens the match evidence at all. As for the rest, my argument remains as follows : Given a reasonable chance a priori chance that the perpetrator is among the 2000 (my 5%) the test results make it very likely ($P(G|E)$ that the perpetrator is indeed among the tested persons, and therefore the one that tested positively. Notice that the argument would not hold if they tested 2000 random people from all over the world. –  Erik May 14 '12 at 12:52
I also calculate 1/80,000. –  Ronald May 14 '12 at 13:23
True. My bad, note that the numerical version above is correct (i.e. 1/80000). Got confused adding up the zeroes in my head. –  Erik May 14 '12 at 13:28
I was also thinking: whereas the first intuition focuses directly on HD: the hypothesis that some one in the database is guilty the latter one focuses on HS: that the accused is actually guilty. What needs to be kept in mind is that these two hypotheses are only equivalent after there was discovered that the database D contains exactly one match. Before making this observation they were not equivalent. I am correct in assuming this? –  Valerie May 14 '12 at 14:43