Tag Archives: expected value

The Expected Value Fallacy in State v. Wright

In State v. Wright, 253 P.3d 838 (Mont. 2011), a woman (whose first name was Sierra) complained that her date raped her. No semen was recovered, but a penile swab from 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. 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?
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.

Id. at 841-42.


The Montana Supreme Court concluded that the source attribution on the part of the prosecutor was the production of confusion and that the analyst’s statements were “internally inconsistent.” However, there is no logical inconsistency in the 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 that the analyst misrepresented the statistics when she stated that “two people in Montana would contribute to this mixture . . . Timothy Wright and [Sierra]” “[a]ccording to the test results” alone. 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, (4) it is practically certain that no one but Sierra contributed the minor profile.



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.


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 DNA 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).