The Expected Value Fallacy in State v. Wright

In State v. Wright, 253 P.3d 838 (Mont. 2011), a woman (identified by the Montana Supreme court as Sierra) complained that her date raped her. No semen was recovered, but a penile swab from Timothy Wright, the man she accused, showed a mixture of DNA. Predictably, the major profile seemed to be the defendant's (presumably coming from his skin cells). The minor profile, however, matched Sierra's, confirming her accusation. The random-match probability was "1 in 467,700 Caucasians" and less for other groups. Id. at 841. The direct examination of the state's DNA analyst included the following colloquy:

Q. When you're determining whether or not [Sierra's] DNA is on that penis, tell me what the language "cannot be excluded" means?
A. So that means that the 16 locations we looked at for a DNA profile was at every of those 16 locations.
Q. So whose DNA is on that penis, that penile swab that you examined at the Lab?
A. Well, it--Timothy Wright and [Sierra] can't be excluded as contributing to that profile.
Q. If you--if you're finding her DNA, how come your conclusion isn't that she's included in the profile? That confuses me.
A. At the Forensic Science Division we don't use the word "included." Instead we use "cannot be excluded." It basically means the same thing. It's just our terminology we use.

Id. The prosecutor pressed on, eliciting the statement that the woman's DNA was present in the penile swab:

Q. Can you explain--let's focus on the Caucasian statistic. Can you explain that statistic to the jury? What's it really mean?
A. So that means that in a population of 467,000 you would expect that one person in that population could be included in this mixture.
Q. All right. How many--what's the population of the state of Montana, do you know?
A. It's approximately a million, just under.
Q. So in this particular scenario we've got a mixture of two DNA's, right?
A. Yes.
Q. Statistically speaking, then, I'm just--I want to make sure I understand you, is there only--are there only two people in the state of Montana that can contribute those particular profiles?
A. Yes. Statistically looking at the state of--or the population of Montana two people in Montana would contribute to this mixture.
Q. Those being whom [sic] according to your test results?
A. According to the test results Timothy Wright and [Sierra].

Id. at 841-42.

The prosecution argued in closing that the woman was included as the contributor and that this meant her DNA was on the defendant. On appeal, the defense contended that the source attribution was a knowingly false statement by the prosecutor.

The Montana Supreme Court concluded that this contention was unfounded, but it also described the analyst's source attribution as "internally inconsistent" with the statements that the defendant "cannot be excluded as a contributor." However, there is no logical inconsistency in this testimony. A test that does not exclude someone includes that person. It may include other people when they are tested--or it may not. If the random match probability is small enough, then the test would be expected to exclude all unrelated people, leaving the defendant (or a twin brother or perhaps a close relative) as the only possible source of the semen.

The actual problem in the case was not that the prosecutor knowingly misrepresented the testimony or deliberately elicited false testimony. It was the analyst erred when she stated that "two people in Montana would contribute to this mixture . . . Timothy Wright and [Sierra]" solely on the basis of the "the test results." The expert apparently reasoned that (1) Montana's population is about 1,000,000; (2) about 500,000 would be women; (3) the exact number of women in Montana with the minor profile is 1 (the random-match probability 1/500,000 times the female population of 500,000); hence, it is practically certain that no one but Sierra contributed the minor profile.

Technically, the quantity 1 is an expected value of a variable X that represents the number of women with the minor profile in a randomly generated population of 500,000 women. Expected values are all around us. If we flip a fair coin twice, the expected number of heads is (1/2) x 2 = 1. But we know quite well that the actual outcomes can vary about the expected value. Thus, the probability that two flips of the coin will produce 2 heads is 1/4. Over many coin flips, this "unexpected value" is expected to occur about 25% of the time. Betting on exactly one head in this situation would produce many losses.

How risky was the analyst's source attribution in Wright? The number of occurrences of the minor profile in the population is analogous to the number of heads that would occur when flipping a very heavily weighted coin 500,000 times. Imagine that we generate many populations of 500,000 profiles from a coin that has a probability of only 1/500,000 on each toss. Some populations will have 0 heads (minor profiles), some will have exactly 1, some will have 2, and so on. The number of heads, X, is approximately a Poisson random variable with mean λ = (1/500,000) x 500,000 = 1. Its probability distribution is f(x; λ) = λ^xe^-λ/x! = e^-1/x! = .368/x! For example f(0; 1) = .368/0! =.368, and f(1; 1) = .368/1! = 0.368.

What is the probability that the analyst is wrong in thinking that there is only 1 woman with the minor profile? Of the many populations we generated with the coin, we can ignore some 36.8% of them--the ones with 0 minor profiles. We can ignore them because the real population has one woman (Sierra)--and possibly more with the minor profile. This leaves 63.2% of the populations to consider.

The analyst will be wrong in asserting X = 1 when we have a population in which X = 2 or more. This situation occurs in every population for which X is not 0 or 1. There are 100% x 36.8% x 36.8% = 26.4% such populations, and all of them are within the 63.2% that apply to this case. Consequently, looking at the DNA evidence in isolation, we conclude that there are 26.4/63.2 = 41.8% possible populations for which the analyst errs in asserting that the minor profile from the swab is Sierra's.

Yet, the jury was informed that the DNA test proved that Sierra's DNA was on the swab. The legal issue on appeal should have been whether this extravagant testimony to which no objection was raised was plain error or offended due process. Cf. McDaniel v. Brown, 130 S. Ct. 665 (2010) (not reaching the due process issue). But regardless of how those issues might be resolved, a well trained DNA analyst should not have testified in this fashion. It is hardly news to the forensic science community that the expected number of DNA profiles in a population must be much less than 1 to strongly support an inference that the profile is unique within that population. See David J. Balding, Weight-of-Evidence for Forensic DNA Profiles 148 (2005) (describing the kind of reasoning employed in Wright as a "uniqueness fallacy"); see also Ian W. Evett & Bruce S. Weir, Interpreting DNA Evidence (1998).

Cross-posted from the Double Helix Law blog. An expanded version is published in D.H. Kaye, The Expected Value Fallacy in State v. Wright, 2011. Jurimetrics 51: 1-8, https://ssrn.com/abstract=1921082.