A litigator who uses probabilistic arguments to his advantage has a powerful and persuasive tool at his disposal. Even an elementary knowledge of probabilistic thinking and knowing when to and when not to employ probabilistic principles can make the difference between convincing a jury of a statement's truth versus allowing them to arrive at their own conclusions independently and often naively. In People v. Simpson, Cochran's infamous "If it does not fit, you must acquit," instruction to a naive jury was used with astounding success, and was laden with epistemological determinism. Little did the jury know that Cochran's statement, if it were truly an honorable and fair practice for determining a trial's outcome, would imply that virtually every single case that has ever been tried must result in the acquittal of the accused. Why? Because nothing ever fits, ever. Statistically (read: logically, rationally), all we can conclude is that the probability of the available evidence is rather low given the assumption of innocence – however, it is never zero, it is never statistically impossible. If the probability of the observed evidence is low given the assumption of innocence, the statistical (again, read "logical") decision is to reject the assumption of innocence, and infer guilt. But again, the probability can never be equal to zero. What this means practically is that there is always a chance that an innocent man is found guilty, and that no amount of, or nature of evidence will ever be enough for the observed evidence (or "data") to perfectly disagree with the assumption of innocence. Cochran, of course, did not tell the jury this. He instead ignored probability altogether, and implicitly swayed a suggestible jury on how to arrive at a verdict decision that is so contrary to longstanding and elementary probabilistic principles in statistical decision theory from which the very "reasonable doubt" criteria is housed.

What the Prosecution Should Have Done

The prosecution should have used a probabilistic argument to their advantage by arguing that the probability of the available evidence (e.g., DNA match, etc.) was extremely low if indeed the accused was an innocent man. A simple analogy and instruction to the jury would have done the trick:

"If a coin is assumed fair, meaning that the probability of heads is equal to that of tails on any given flip, then what am I to think of the coin after flipping it one hundred times and getting 100 successive heads? The probability of data like this (i.e., the 100 heads), when assessed statistically, is so low that we end up casting doubt that the coin is fair in the first place. In other words, our witness of a series of flips like this is an extremely rare event if the coin were actually fair. Consequently, we end up rejecting the assumption of "fairness" and conclude instead that the coin is biased."

Drawing on the above coin analogy, it would have been so easy for the prosecution to link this type of reasoning to the task of the jury:

"Likewise, if Mr. Simpson is indeed innocent, then the probability of having this kind of surmounting evidence against him (e.g., DNA match, etc.) is astronomically low to be virtually impossible – rationally, rejecting the assumption of innocence is a no-brainer, just as rejecting the assumption that the coin is fair is a no-brainer. In other words, our witness of a series of events as has been presented to the court is an extremely rare event if the accused were actually innocent. Consequently, he must be guilty within a reasonable doubt."

However, the jury was never presented with this type of probabilistic framework, and so they were easily swayed by Cochran's deceptive and deterministic (and logically false) non-probabilistic argument. Had the prosecution encouraged the jurors to think and decide using statistical reasoning, they would have been rationally obligated to convict Mr. Simpson, just as easily as they would have found the coin to be unfair. Teaching the jury a few basic probabilistic principles would have done wonders in instructing them on how they were to decide the verdict.

The Lesson: Probability is Your Ace in Court

The take-home message here is that by either introducing or avoiding probabilistic principles into an argument, a trial lawyer can set the criteria by which a juror will decide. As in the O.J. Simpson case, if probability works against you, one can deliberately deny probabilistic ideas in the verdict instruction as Cochran did, and adopt a more deterministic approach. If, however, probability works for you, then one can easily illustrate using an elementary coin-flip example how it is the jury's job to assess the probability of the available evidence given the assumption under test, that of innocence of the accused. If that probability is quite low, one has a strong statistical argument to conclude the assumption of innocence must be doubted, and hence rejected.

Always present and interpret evidence with a keen awareness of how you are or how you are not using probabilistic reasoning in your argument to a jury. It can make the difference between winning and losing a case.

*Daniel J. Denis, Ph.D.** is * *Associate Professor of Quantitative & Statistical Psychology at the University of Montana, where he teaches advanced courses in decision-based statistical modeling, and heads the Data & Decision Lab in the Department of Psychology. Dr. Denis specializes in the teaching of statistical and mathematical concepts through a combination of analytical, practical, and historical analyses, as well as provides statistical and data-analytic services to clients in psychology, law, and health sciences.* *He can also be reached through the **Data & Decision Lab**.*

**We asked a trial consultant to respond to this article and Ken Broda-Bahm offers his thoughts below. **

**Ken Broda-Bahm responds to: **

**Ken Broda-Bahm is a senior litigation consultant with Persuasion Strategies in Denver, Colorado.**

Dr. Denis makes the argument that litigators should express arguments more accurately using probabilistic terms. It is hard to argue against the use of greater logical rigor in delivering and interpreting arguments in court. At the same time, however, it is the lawyer's responsibility to understand the reactions to those arguments from a jury that may lack even a basic knowledge of logical probability. The question that I would like to raise is whether the choice to portray innocence or guilt in explicitly probabilistic terms is at war with either the default human tendencies toward handling probabilities, or the practical demands of advocacy.

Most people are likely to be naive thinkers when it comes to probability. We can understand the general concepts, and we are fine with broad constructs like "more likely," "less likely," and "highly unlikely." Yet, outside of realms of mathematics or coin flips, it seems unlikely that most people would be able to logically quantify chance. To take Dr. Denis' example of the preferred formulation, the prosecutor can say that the evidence against Mr. Simpson is equivalent to a coin hitting heads in 100 out of 100 flips, but is it? Does the courtroom spectacle of Mr. Simpson failing to get the glove onto his hand take that number down to 80? Possibly, 50? Or maybe to 25? If that seems like a square peg trying to fit into a round hole, it is because it is an attempt to apply mathematical rules to a matter of fuzzy human judgment. Jurors want to think about Mr. Simpson as guilty or innocent, and even for those who understand and are willing to apply a "beyond a reasonable doubt" standard, it is unlikely that this standard could ever be put into mathematical terms.

Attempts to quantify the chances of innocence or guilt can come at a cost to the credibility of the advocate. For example, I suspect that no prosecutor would want to stand before a jury and state that the accused is "probably" guilty, or worse yet, to compare their case to a series of coin flips. Invoking the real element of fundamental uncertainty may be logical but it seems unlikely to persuade. To the prosecutor, and to the world-view that the prosecutor invites the jurors into, the accused is guilty, plain and simple. Juries can, should, and often do rely on burden of proof and comparative probability to resolve an impasse created by the two different narratives, but when the advocates themselves on either side make their arguments explicitly probable by invoking burden of proof, it is generally a sign that their case is not going well.

It is likely that Mr. Cochran expressed his argument in definite rather than probable terms simply because it made for a more effective argument that way. If you grant the defense the benefit of the doubt, realizing that is hard for many to do on People v. Simpson, Mr. Cochran's argument focused on the single anomalous fact that casts into doubt the entire body of evidence from the prosecution. Rather than expressing it as a degree of doubt introduced by the single fact, it is better and arguably more accurate to express it based on its bottom line: if the killer had smaller hands than Mr. Simpson, then Mr. Simpson is not the killer. We now know that there were many reasons other than possible innocence why the gloves "didn't fit": e.g., shrinkage while in evidence storage (later in the trial, Mr. Simpson tried on a new glove of the same style and it fit perfectly). The fact that the prosecution failed with this jury to see the failed fitting as a logical ruse shows perhaps that jurors treated the glove less as a logical refutation and more as a metaphor for the many argued facets of the prosecution's case that didn't fit perfectly. Dr. Denis' point that it will never fit perfectly is well taken, but the solution isn't to treat jurors as machines capable of calculating probability, but to treat them as narrative reasoners who assess the rough comparative likelihood of two different stories.

**Daniel Denis replies:**

In his reply to my article, Ken Broda-Bahm argues that a prosecutor would be at a disadvantage to introduce the concept of probability into a juror's decision-making process. Though he is absolutely correct that most jurors (and judges) have some difficulty with assimilating probabilistic arguments, we do appear to be in agreement that probabilistic thinking is in the brainwork of a juror regardless of whether or not he or she is aware of it. What is the "comparative likelihood" task Broda-Bahm alludes to other than a job of assigning probabilities? Further, how likely does a story have to be before I, the juror, send a man to jail? How likely does that same story have to be before I send the same man to death row? It is a question of chance, and one cannot escape probabilistic thinking in arriving at a legal verdict, assuming it be a rational one.

To clarify, I am not arguing that attorneys introduce probability concepts into every case, nor am I suggesting that we ask jurors to perform complicated and seemingly trivial academic mathematical computations of real empirical probabilities. Far from it. My point rather is that there are instances where the simple concept of probability can be effectively used as a powerful weapon of persuasion and instruction. At minimum, it behooves counsel to be aware of when probability concepts are, or are not put in play by their opponent, so they may counter with greater success.

Mr. Cochran essentially told the jury that if a piece of evidence does not fit perfectly into the mental imagery of a guilty man, then regardless of all the other pieces that do fit (and fit quite well), the accused must be set free. The suggestible jury was likely unaware that his instruction was pure manipulative nonsense, and the least the prosecution could have done was to anticipate with their own preparatory lesson on how, logically, they were to arrive at a rational verdict. Teaching a jury how to make rational decisions, which necessarily implies the instruction of very basic and elementary probabilistic reasoning, may be a worthwhile investment in your case. If jurors were better educated on how to assess evidence as inputs to a decision, perhaps they would make better ones, instead of basing them on emotional, political, and other heavily biased breeding grounds that are the faithful servants of irrational decision-making.

Citation for this article: *The Jury Expert*, 2010, *22*(4), 29-32.

"[…I]t would have been worth a shot – imagine that, when all else fails, just ask the jury to be rational!" This is a very poignant statement. I am also rather persuaded by Mr. Barnes' suggestion that probabilistic arguments could fare better in civil trials, where what matters is the preponderance of evidence. A long literature and a long list of cases shows that criminal courts have been unsuccessful in quantifying what the standard for reasonable doubt should be, and that in fact many judges have been reluctant even to try, or to tolerate attempts in this direction. I am reminded of math-popularizer John Allen Paulos's quote: "There are times when not being quantitative is a kind of false piety which can only make obscure and thus more difficult the choices we must make."

I believe that statistical and probabilistic reasoning is important, as it helps me to analyze situations. I use it regularly to think about all sorts of matters. My own liking of statistics and probability does *not* lead me to recommend to attorneys that they present probabilistic thinking to jurors. I often recommend other approaches than educating jurors in statistical thinking or probabilities because of my beliefs about what I think jurors will find most persuasive.

I agree that there is a probabilistic fallacy in "If it doesn't fit, you must acquit". My approach to handling that fallacy would be a bit different. I have found an effective persuasive response to such mantras or quips is to "counter-quip": "If he can get the gloves on at all, you know he's guilty. Who brings *new* gloves to a murder?". Such counter-quips recognize the underlying issues without going into the statistical and probabilistic underpinnings, and jurors are likely to remember the counter-quips. And, after the counter-quip, if it is necessary to educate jurors about baseline probabilities, improbabilities, or statistical realities, we do so, but only to the degree that is necessary. I do not believe this education would have been necessary in the OJ Simpson case in order to address Mr. Cochran's statement.

My experience has been that jurors stay with us when we use "counter-quips" and jurors either have difficulty understanding or are inattentive when we try more educational approaches absent such quips. My experience is also that the point of these counter-quips stays with jurors while the point of a more logical and analytical response might not.

Persuasion research often finds that what is logical is not necessarily the most persuasive response. Persuasion research reports, for example, that metaphors and analogies are more persuasive than statistics, probabilities and facts, even when such devices as figurative metaphors carry no "factual" value. We understand the metaphor, we can remember the metaphor, and the metaphor (if done well) is persuasive.

This is not to say that I advocate avoiding statistical or probabilistic education of jurors when it is necessary. I recommend this education when I believe it is necessary, but I recommend it only to the degree that it is necessary and presented only in ways that jurors find easiest to understand and remember. This approach fits with my basic mantra: "Reduce our persuasive burdens, increase the other side's persuasive burdens." Taking on statistical or probabilistic education of a jury is usually a burden, and so I approach it when necessary, to the degree necessary, and in the most persuasive ways possible.

Perhaps probabilistic arguments would work better in civil court where the standard of proof is preponderance of the evidence, i.e., 51% more likely. Civil litigation is more often a close call between two relatively rational viewpoints – something close to "even odds" of winning or losing. Disputes farther away from this middle ground fall out of the system through settlement or summary judgement. So, in these closer cases, presentations to the jury about which scenario is "most likely" through the use of probabilistic arguments may be more well received and effective.

I don't think that a trial of the nature of People v Simpson is a good example to use for this article. Too many externalities come to bear to allow us to test, through this mental exercise, the relative merits of Cochran's and Denis's methods.

Thank you, Mr. Barnes, for your comment. I would say that the Simpson trial was a suitable case to instruct the jury on how to make a rational decision for the very reason you mention – there were so many competing motives that ended up going into the verdict. Here was a case where true reasoning and rational behavior was required. Most reasonable people would have found Simpson guilty, and had the jury understood their task, and been taught a few principles on how to make a coherent decision that is void of the emotional/political push and shove, it would have been an easy conviction. My point is that the prosecution might have been better off simply teaching the jury what constitutes a *good decision,* one based on logic and rationality rather than manipulative arguments. Keep in mind that in this I (and virtually all rational decision theorists) assume there is an "optimal" way of making choices, which is what is prescribed by decision theory. What makes decision theory "optimal" you might ask? Ultimately, it's a similar reason why we can agree that the shortest distance between two points is a straight line – the methods prescribed by decision and statistical theory are the most "rational" methods we know of on how to make good decisions (and centuries of "reasonable" mathematics back it up). So in sum, it's exactly in this type of controversial media frenzy case where rationality is needed, to cut through the hype, and had someone on the prosecution approached them and basically said, "Look, this isn't that complicated, almost any rational person would convict, and here's why [enter examples of rational decision-making], we're just asking you to make a rational decision." It seems to me that if the jury did consist of even semi-rational people, they would have had to convict. Of course, they probably acquitted for numerous reasons (political, racial, emotional), but my point is only that they should have been instructed on some kind of framework to make their decision correctly. Does a jury have any framework for understanding what a "reasonable doubt" really is? "Reasonable doubt" is best (or even only) explained by probability examples, and if juries understood these ideas, I think they'd better understand their task as a decision-maker. Regarding the Simpson case, what Cochran asked the jury to do (if it does not fit, you must acquit) was not rational. Had the prosecution instructed them on what WAS rational (and the principles behind it), the jury might have seen through Cochran's statement. True, if the jury was too emotionally or politically involved at that point to give rationality a chance, then that would have probably made a tutorial on rational thinking irrelevant, regardless on how well it was taught. But, I still think it would have been worth a shot – imagine that, when all else fails, just ask the jury to be rational!

@DocRobPhD23 @ASTCNow @APLS41 @SocialPsych @Harvard_Law @TrialConsultant He wrote for TJE in 2010. https://t.co/quGMk6jTni

RT @TheJuryExpert: @DocRobPhD23 @ASTCNow @APLS41 @SocialPsych @Harvard_Law @TrialConsultant He wrote for TJE in 2010. https://t.co/quGMk6jT…