What’s the Mathematically Optimum Jury Size? Is it 12, Six, or One?

The state of the universe.
April 25 2013 11:40 AM

What’s the Best Jury Size?

The Supreme Court rejected math before, but it’s now possible to calculate the jurors and margins you need for justice.

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Another problem with the Condorcet model is the assumption that jurors make the “right” decision a certain fixed percent of the time. Who is to say what is right? How can we ever measure the competence of a juror?

In a very provocative 1992 paper, George Thomas, a law professor at Rutgers University, and his student Barry Pollack, now a partner at Pollack Solomon Duffy LLP in Boston, argued that the function of a jury is to serve as a proxy for society. In ancient Greece every citizen of the polis served on the jury. In the modern world this is impractical, so we settle for juries of 12.

According to Thomas and Pollack, then, the most objective measure of a jury’s success is whether it agrees with what society would have decided. In his lecture, Suzuki phrased it more cynically: The function of a jury is to preserve the appearance of justice. Consider the fact that the most notorious judicial outcomes of recent years—the O. J. Simpson case, the Casey Anthony trial—are exactly the ones where society disagreed with the jury’s decision. Or, looking at it the opposite way, the jury failed to do its job of agreeing with the rest of us!


Whether or not you subscribe to this definition of a jury’s purpose, Thomas and Pollack’s model does make it possible to determine mathematically how likely such disagreements are. If you pick 12 people at random, how likely is it that they will disagree unanimously with the majority of society? Not very likely. How likely is it that they will disagree with society by a 9-to-3 majority? Thomas and Pollack crunched the numbers, and Suzuki recrunched them. And they found a surprising consistency. For every margin that the Supreme Court has allowed to stand (6- to 0, 10 to 2, 9 to 3), the probability of a disagreement between society and the jury is less than 1.5 percent. And for every margin that the Supreme Court has ruled unconstitutional (5 to 1, 5 to 0), the probability of disagreement is greater than 1.5 percent. Thus, without realizing it, the Supreme Court has consistently held that there should be less than a 1-60 chance that the jury will disagree with society. Judicial hunch meets mathematical rigor!

However, the Supreme Court should not pat itself on the back too hard. Suzuki pointed out that nonunanimous juries fail the Thomas-Pollack test if some of the jurors are biased in favor of conviction. A 12-member jury with a unanimity requirement, he showed, can absorb up to five biased jurors and still meet the 1.5 percent threshold. By contrast, if the required margin for conviction is only 10 to 2, then a single biased member is enough for the jury to make the wrong decision (from society’s point of view) more than 2 percent of the time. Six-member juries fare even worse. Even with a unanimous vote required to convict, one biased juror will cause the jury to make the wrong decision more than 5 percent of the time. Both 6-to-0 and 10-to-2 convictions, Suzuki concluded, should be held unconstitutional. “The Supreme Court should reconsider its decisions and prohibit six-person juries,” he said.

Will the Supreme Court pay attention to mathematics this time? Given its record, the answer seems almost certain to be no. “The Supreme Court is remarkably uninterested in statistical research,” says Thomas. “The jury is, according to the court, a black box that deserves to remain black as long as the verdicts stay within the parameters the court has set out.”

However, the issue is not completely dead, because the Supreme Court continues to receive appeals from Louisiana and Oregon, including Alonso Herrera’s. (The court was interested enough in that case to request a brief from the state of Oregon, but it ultimately declined to hear his appeal.)

In addition, there are very good legal (as opposed to mathematical) reasons to reconsider the decisions of the 1970s. The discrepancy between federal and state law is perhaps the most compelling one. The Fourteenth Amendment expressly extended Bill of Rights protections (such as the jury trial) to the states. In Apodaca v. Oregon, eight of the nine Supreme Court justices agreed that federal and state laws should require the same quota for a conviction. The only trouble was that the justices didn’t agree what the quota should be. Four thought it should be unanimous, and four thought it should be nonunanimous. Justice Powell—yes, him again—broke the deadlock. He was the only justice who believed federal juries should be unanimous but states should be allowed to “experiment” with nonunanimous juries. Perhaps it is time to declare the experiment over.

Correction, April 26, 2013: This article misstated that Louisiana requires a vote of 9 to 3 to convict in noncapital cases. This was true at the time of the Supreme Court cases, but the state has since changed its threshold to 10 to 2. (Return to corrected sentence.)

Dana Mackenzie is a freelance mathematics and science writer based in Santa Cruz, Calif. He is the author of The Universe in Zero Words and The Big Splat, or How Our Moon Came to Be. He also writes a popular chess blog.