Explaining Probability? Use Frequencies Rather Than Percentages

Holland & Hart - Persuasion Strategies

Jurors and judges sometimes need to understand testimony regarding probability. For a criminal jury, maybe that probability relates to the chances of a false-positive on DNA identification. In a products case, maybe it concerns a failure rate. In an employment discrimination class action, it may relate to differing hiring percentages. And in a legacy contamination case, perhaps it relates to the risks of the future spread of a toxin. In each of these situations, the judge or jury will need to understand and apply the statistics in a realistic manner. That isn’t always easy, and often won’t be as easy as the highly-analytical attorneys and trained experts assume that it will be.

Some recent research (Weber, Binder & Krauss, 2018), however, points to one factor that will make it more understandable: Instead of talking about percentages or proportions, use natural frequencies. In other words, instead of saying “ten percent,” say “one in ten,” or instead of saying “twelve percent,” say “three in twenty-five.” Why? After all, they mean exactly the same thing and one might assume that the mind is easily able to translate from one expression to the other. Assuming that, however, would be a bias based on one’s own education or analytical style. Research has long shown that when the explanation is based on frequencies rather than percentages, you’ll have a much greater chance that your audience will understand and correctly apply what you are saying. In this post, I will take a brief look at why that is, and discuss the practicalities of how to frame probabilities in the courtroom.

The Natural Frequency Advantage

There seems to be a built-in tendency to see probabilities in terms of percentages. As explained in a recent ScienceDaily release, lead author Patrick Weber explains, “Even though natural frequencies are much easier to understand, people are more familiar with probabilities represented by percentages because of their education.” The percentage might be more familiar and seem to be more precise, but it is not more understandable. Weber continues, “A recent meta-analysis showed the vast majority of people have difficulties solving a task presented in probability format.” That research showed that when the task was explained using natural frequencies instead of probabilities, performance rates increased six-fold, from 4 percent to 24 percent.

One version of the test is as follows:

The percentage version: The probability of being addicted to heroin is 0.01 percent for a person randomly picked from a population (base rate). If a randomly-picked person from this population is addicted to heroin, the probability is 100 percent that he or she will have fresh needle pricks (sensitivity). If a randomly-picked person from this population is not addicted to heroin, the probability is 0.19 percent that he or she will still have fresh needle pricks (false alarm rate). What is the probability that a randomly- picked person with fresh needle pricks is addicted to heroin (posterior probability)?

The natural frequency version: 10 out of 100,000 people from a given population are addicted to heroin. 10 out of 10 people who are addicted to heroin will have fresh needle pricks. 190 out of 100,000 people who are not addicted to heroin will nevertheless have fresh needle pricks. What percentage of the people with fresh needle pricks is addicted to heroin?

The answer, for both versions, is just 5 percent or 1 in 20. That number might initially seem very low, but that is only because people tend to inflate the baseline based on the “10 out of 10” heroin addicts having fresh needle pricks. The natural frequencies approach fares better because people can simply add it up using the same real base of 100,000.

The Reason Understanding is Still Low

If you had trouble working through the examples, you are in good company. Even with the more comprehensible frequencies version, fewer than one in four research participants will accurately solve the task. When people fail, according to the authors, it is because they persist in converting even the frequencies back to percentages, and make errors as a result. The theory they tested is that people tend to fail at understanding the statistics because they prefer the complicated percentage-based solution over the more intuitive frequencies, possibly due to a fixed mindset they acquired in school.

The researchers in this case are dealing with German University students, and that likely makes a difference in this kind of study, particularly since it seems German students are learning statistics in high school. Still it is noteworthy, that the failure rate is so high, and the reason — an ingrained habit of thinking that is tough to set aside — is generalizable. It is a useful reminder that people bring their own perceptions and preferences when they set out to solve a problem.

What Courtroom Persuaders Should Do About it

Number one, don’t take it for granted that your fact-finders will understand what you need them to understand. The right approach for teaching the right probability science will vary from case to case. But there are a few rules of thumb:

  • Teach. Make sure you don’t just present the evidence, but instead explore and test out ways to boil it down, to make it simple and sticky.
  • Use multiple methods. Some jurors might understand it as a percentage, others as a frequency, and still others as an example or a story. Use as many methods as you can, and make sure they’re consistent.
  • Use visual aids. Particularly when representing proportions, hearing it is one thing, and seeing is another. “One in ten” as a visual representation is going to be more immediately meaningful and memorable.


Weber, P., Binder, K., & Krauss, S. (2018). Why can only 24% solve Bayesian reasoning problems in natural frequencies: Frequency phobia in spite of probability blindness. Frontiers in Psychology, 9, 1833.

Image credit: 123rf.com, used under license, editedRelated Posts Plugin for WordPress, Blogger...


DISCLAIMER: Because of the generality of this update, the information provided herein may not be applicable in all situations and should not be acted upon without specific legal advice based on particular situations.

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