Misleading DNA evidence and the current damaged winning lottery ticket story

Norman Fenton, 28 January 2016

This post is primarily about how DNA match evidence is often presented in a way that is highly misleading (it is an important issue in an ongoing case I'm involved with). But in order to illustrate the point it turns out that we can use a simple analogy based loosely on the current lottery story that is getting a lot of media attention in the UK. This concerns an unverified £33 million winning ticket from a recent draw. About 200 people are claiming to have bought the (single) winning ticket but, until today*, none had actually provided proof of possessing such a ticket. The claim of one - Miss Susan Hinte - is the one that has grabbed media attention because she has produced a ticket in which key identifying information cannot be read because, she claims, the ticket was put through a washing machine.

But first let's look at the DNA issue, which is concerned with the following generic problem:

• The prosecution claims that defendant Joe was at the crime scene. This hypothesis is denoted as Hp.
• A tiny trace of DNA from the crime scene has been analysed and found to match the profile of Joe. This evidence (of the match) is denoted E. 

Typically the defence will argue that Joe was not at the crime scene and that any DNA matching Joe - especially as it was a tiny trace - got there through secondary transfer or other means. So the defence hypothesis Hd is simply the negation of Hp.

The DNA experts have correctly recognised that, in determining the probative value of the evidence E,  they have to use the ‘likelihood ratio’ approach [1]. This means they have to consider both of the following probabilities:

1. The probability that E is the result of the prosecution hypothesis Hp being true  - formally we write this as P(E given Hp)
2. The probability that E is the result of the defence hypothesis Hd being false  -  formally we write this as P(E given Hd) 

If probability 1 is greater than probability 2 then the evidence E supports  Hp over Hd and vice versa. The likelihood ratio is simply 1 divided by 2 and provides a simple and compelling measure of probative value of evidence. If the ratio is greater than one the evidence E supports Hp, with higher values indicating stronger support. If the ratio is less than one the evidence E supports Hd, with smaller values indicating stronger support. However, for reasons explained in [1], this whole notion of probative value is not meaningful if the defence hypothesis Hd is not the negation of the prosecution hypothesis Hp. One of the common errors made by DNA experts is to replace Hd with a different hypothesis, namely Hd':  "DNA from Joe got there by secondary transfer".  In this case Hd' excludes other possibilities of observing E even though Joe was not at the crime scene (such as errors or contamination during the DNA testing, or the DNA belonging to a different person with the same profile etc) and is not even mutually exclusive to Hp since Joe may have been at the crime scene even though the trace sample was there through secondary transfer. But, while this common error is serious, it is not the real concern I wish to raise here. In fact, let's suppose that no such error is made and that the expert considers the correct Hd.
. 
The real concern is how a jury member reacts when the DNA expert now makes the following assertions:

1. “The findings are what I would have expected if Hp were true.” i.e. P(E given Hp) is very high
2. “The probability of the findings are considerably more likely to have been the result of Hp rather than Hd”  i.e. P(E given Hp) is much higher than P(E given Hd)

Notwithstanding the unnecessary redundancy of statement 1, these assertions sound very important and suggest very strong support for the prosecution hypothesis, especially as most people would already have assumed (wrongly) that the DNA 'match' means the trace certainly belongs to Joe.

But to demonstrate how misleading they are I will return now to the lottery example. For simplicity I will assume the old 6-ball lottery with 49 numbers. Suppose the winning numbers were:
1, 7, 21, 28, 40, 46

Mrs Smith has a damaged ticket that she claims has the winning numbers. The evidence E is that the first number (which is the only number clearly visible) is 1.

Our hypotheses are:

• Hp: “Mrs Smith's ticket is the winning ticket”
• Hd: “Mrs Smith ticket is not the winning ticket”

In this case we know the following:

• P(E given Hp) = 1  (it is certain that the first number on the ticket would be 1 if it was the winning ticket)
• P(E given Hd) is 0.122 (this is the proportion of non-winning tickets that have 1 as the first number) 

So we could certainly make exactly the same assertions in this case as the DNA experts above:

1. “The findings are what I would have expected if Hp were true.” (since the probability of E given Hp is 1)
2. “The probability of the findings are considerably more likely to have been the result of Hp rather than Hd” (since 1 is considerably greater than 0.122).

However, despite these (correct) assertions it is almost certain that Hd rather than Hp is true - Mrs Smith's ticket is not the winning ticket. In fact, the probability of Hp being true is less than one in 1.7 million (because there are over 1.7 million non-winning combinations in which the first number is 1).

So what is the moral of this story? The likelihood ratio of the evidence might often suggest the evidence is highly probative in favour of one of the hypotheses, but if the prior probability of the alternative hypothesis was much higher to start with then the evidence will not ‘overturn’ the prior belief in favour of the alternative.

Lay people ignore this in connection to DNA evidence. Because the random match probability associated with a DNA match is typically less than one in a billion, the very fact that the evidence E is a "DNA match" already puts into their mind the notion that this 'must tie the defendant to the crime scene'. But the random match probability is almost irrelevant in this case - it only accounts for a tiny proportion of P(E given Hp). Lay people can also easily be tricked into believing that the (redundant) assertion 1 “The findings are what I would have expected if Hp were true” provides additional weight to assertion 2.

Unfortunately, this type of evidence is increasingly prejudicing juries and, I believe, leading to serious miscarriages of justice.

*The real winner has now been found, and since their ticket was not damaged it can not have been Miss Hinte

[1] Fenton, N. E., D. Berger, D. Lagnado,  M. Neil and A. Hsu, (2014). "When ‘neutral’ evidence still has probative value (with implications from the Barry George Case)",  Science and Justice, 54(4), 274-287 http://dx.doi.org/10.1016/j.scijus.2013.07.002. (pre-publication draft here)