DNA Test Results: Probability vs. Fallacy

The State of North Carolina goes to trial against Donnie Defendant, who is alleged to be the infamous “Tarheel State Killer” and charged with committing a series of brutal assaults and murders several decades ago. The state’s case depends heavily on matching DNA evidence from the crime scene to a sample of DNA taken off a cigarette butt discarded by Donnie. At trial, Special Agent Wanda Witness testifies as an expert in forensic DNA analysis for the state. After explaining the science behind PCR, STR, loci, and markers, Wanda opines that Donnie’s DNA is indeed a match to the DNA recovered from the crime scene.

Sounds like good news for the state, but what exactly does a “match” mean? And how may the significance or statistical probability of that “match” be expressed to the jury? It’s an important question, because what might sound like two similar ways of expressing the same probability can have dramatically different meanings – and possibly even be considered error on appeal.

Testimony and Argument at Trial

Continuing her direct examination, Wanda testifies that the “random match probability” in this case is calculated to be 1 in 10,000,000. She explains that this means if one were to go out and select an unrelated person completely at random, and then compare that person’s DNA to the DNA sample taken from the crime scene, the likelihood that those two samples would happen to be a match is only one in ten million.

Wow. Strong evidence. Paula Prosecutor thinks so too, and she makes it the central focus of her closing argument: “Reasonable doubt? Ladies and gentlemen, gimme a break – the DNA evidence tells you that the odds are literally ten million to one that it was anybody BUT him!”

Wait a minute – is that right? Paula probably thinks she’s just saying the same thing in a different way (one in ten million vs. ten million to one), but is she?

Unfortunately for the state, no – it’s not the same. And in some cases, it could be viewed as error on appeal.

The Prosecutor’s Fallacy

Although this mistake of logic can just as easily be made by judges and jurors (or even by the testifying witness herself), the phenomenon is usually described as the “prosecutor’s fallacy.” Jessie Smith has summarized this issue in her Benchbook entry on Expert Testimony (p.26), but let’s dig a little deeper into the rationale behind it.

The key to understanding the fallacy is to recognize that we are talking about two fundamentally different questions:

  • Assuming the person is innocent, what is the likelihood that his DNA would be a match to the sample from the crime scene?
  • Assuming that the DNA is a match, what is the likelihood that the person is innocent?

The prosecutor’s fallacy takes the answer to the first question and uses it is an answer to the second question. That’s flawed logic, because you’re actually talking about two very different concepts: random match probability vs. source probability. The U.S. Supreme Court explained it this way in McDaniel v. Brown, 558 U.S. 120 (2010):

[I]f a juror is told the probability a member of the general population would share the same DNA is 1 in 10,000 (random match probability), and he takes that to mean there is only a 1 in 10,000 chance that someone other than the defendant is the source of the DNA found at the crime scene (source probability), then he has succumbed to the prosecutor’s fallacy. It is further error to equate source probability with probability of guilt, unless there is no explanation other than guilt for a person to be the source of crime-scene DNA. This faulty reasoning may result in an erroneous statement that, based on a random match probability of 1 in 10,000, there is a 0.01% chance the defendant is innocent or a 99.99% chance the defendant is guilty.

Clear as mud? Stay with me.

The Defender’s Fallacy

I find it easier to understand why it’s a fallacy by looking at it from the other side. That’s right, there is also a “defender’s fallacy,” and one version of it could go something like this: you just said the odds of a random match are 1 in 10 million? Well, the current U.S. population is a little over 320 million, which means there must be at least 32 people in this country who could match this DNA profile. So according to the DNA, the odds are actually 31 to 1 that the defendant is NOT the person who left this DNA at the scene. In other words, there is a 97% chance he’s not guilty!

Obviously that’s not right. For one thing, it ignores the fact that some of those 32 people have to be excluded as possible sources, like infants and people who live in Missouri. But more importantly, this statement of “probability” fails to take into account all other factors in this case which point to this defendant, like motive, opportunity, corroborating evidence, eyewitness identification, or whatever it may be. If, in addition to the DNA evidence, there were also a dozen eyewitnesses who personally knew the defendant and saw the crime happen, plus a video of the entire incident as it occurred, and a signed confession, would it make any sense to say there is only a 3% chance the defendant is guilty? Of course not.

But by the same rationale, the state can’t say that there is a 99.99999% chance the defendant is the source (and is therefore guilty) based solely on a DNA match, because the actual likelihood of his guilt depends on weighing all the factors in the case. For example, even if the state has a DNA match from the crime scene, the odds of that defendant being the perpetrator would be dramatically lower if he had a solid alibi and could prove that he was in the hospital at the time of the murder – which is exactly what happened in this fascinating case from California.

So Will Donnie’s Case Be Reversed on Appeal?

Probably not.

Wanda Witness was pretty careful in how she explained random match probability during her direct examination, and she didn’t get it mixed up with source probability. In fact, Wanda could have gone even further and expressed her opinion that it would be “scientifically unreasonable” to conclude that anyone other than the defendant is the source of the DNA from the crime scene, as long as she refrained from directly equating random match probability to source probability. See State v. Oaks, 232 N.C. App. 338 (2014); State v. Neal, 216 N.C. App. 183 (2011) (unpublished); cf. State v. Ragland, 226 N.C. App. 547 (2013).

As for Paula, she was clearly mistaken in the way she rephrased the probability evidence during her closing, but it’s possible the Court of Appeals would allow her a little latitude, since counsel are permitted to draw inferences from the evidence during arguments to the jury. See State v. Spinks, 244 N.C. App. 345 (2015) (unpublished) (no error where prosecutor argued that “because the random match probability was one out of more people than are on the planet, defendant must be the source of the DNA profile found at the crime scene” – court found that “the prosecutor’s comment was not beyond the latitude allowed during closing argument”).

The greatest risk of error seems to arise when the prosecutor asks an expert witness to provide (or agree with) a specific number related to source probability on direct examination. Watch out for those questions that start with words like “so, basically what you’re saying here is…” and end with the witness saying “yes, I guess you could say that.” See McDaniel v. Brown, 558 U.S. 120 (2010); State v. Whitley, __ N.C. App. __, 790 S.E.2d 755 (2016) (unpublished).

For prosecutors, I think the main takeaway is that they need to carefully plan their probability questions in advance, and be mindful of what the witness can and can’t say on the issue. Defense attorneys may also want to pay close attention to any source probability questions, since several of the cases cited above turned on the fact that even if the testimony was error, the issue was not properly preserved.

2 thoughts on “DNA Test Results: Probability vs. Fallacy”

  1. Worst of all, I’m also a “Tarheel” born and a “Tarheel” bred… can’t believe that mistake slipped through. Thanks for catching it.


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