Mathematics shows up in law wherever a dispute turns on a number: the odds that a DNA sample came from someone other than the defendant, the probability that a ledger was fabricated, the shape of a legislative district, the risk that a defendant will reoffend, and the lump sum that fairly replaces decades of lost wages. Understanding how math is used in law means understanding both the techniques themselves and the legal rules that decide when a judge or jury is allowed to hear them.
Probability in Forensic Evidence
Forensic analysts routinely combine individual probabilities to estimate how rare a particular combination of traits is. The method, called the product rule, treats each characteristic as independent and multiplies the frequencies together. A DNA profile that matches at several genetic markers, each with its own population frequency, produces a combined figure like “one in a billion” meant to tell the jury how unlikely a coincidental match would be.
The rule only holds when the variables are actually independent. The 1968 California case People v. Collins is the standard warning. A prosecutor had a mathematician assign probabilities to traits the robbers allegedly shared, including an interracial couple, a yellow car, and a woman with a ponytail, then multiplied them to tell the jury the odds of another couple matching were about one in twelve million. The California Supreme Court reversed, finding that the prosecution never proved the traits were independent and never established a reliable statistical foundation for the numbers themselves.
Collins did not ban statistical evidence. It required that mathematical testimony rest on verified assumptions rather than arbitrary probability assignments. A jury that hears “one in twelve million” tends to treat the figure as certainty, and when the figure is built on guesswork, the conviction rests on an illusion of precision.
The Prosecutor’s Fallacy
Even sound statistics can be presented in a way that misleads. The prosecutor’s fallacy confuses the probability of the evidence given innocence with the probability of innocence given the evidence. If a forensic test produces a false positive one percent of the time, a prosecutor committing the fallacy will tell the jury there is a 99 percent chance the defendant is guilty. That is not what the number means. One percent describes how often the test would wrongly match an innocent person, not the overall likelihood of guilt, which also depends on how many people could have been the source.
The British case of Sally Clark shows the cost. Clark was convicted in 1999 of murdering her two infant sons after a pediatrician testified that the probability of two sudden infant deaths in the same family was roughly one in 73 million. That figure came from squaring the probability of a single cot death, about one in 8,543 for a family with Clark’s demographic profile, as though the two deaths were fully independent. Later analysis showed they were not: families with one sudden infant death face a significantly elevated risk of a second. Clark spent more than three years in prison before her conviction was overturned on appeal.
These errors persist because probability is counterintuitive and juries are not trained to tell apart “the chance of seeing this evidence if the defendant is innocent” from “the chance the defendant is innocent given this evidence.” English courts have gone so far as to restrict formal Bayesian probability calculations in criminal trials, reasoning that the framework confuses juries more than it helps them.
How Courts Screen Mathematical Testimony
Federal courts have a formal filter for keeping bad math out of the courtroom. In Daubert v. Merrell Dow Pharmaceuticals (1993), the Supreme Court held that trial judges must act as gatekeepers, screening expert testimony for scientific reliability and relevance before a jury hears any of it. Judges weigh factors including whether the method has been tested, whether it has undergone peer review, its known error rate, whether standards govern its use, and whether the scientific community broadly accepts it.
Federal Rule of Evidence 702 codifies that gatekeeping role. A qualified expert may testify only if the proponent shows the court it is more likely than not that the testimony rests on adequate facts, uses reliable methods, and applies those methods reliably to the case. A 2023 amendment added the explicit “more likely than not” preponderance standard after years of inconsistent thresholds across courts.
In practice, a forensic statistician who wants to present a DNA probability calculation must first survive a pretrial hearing. The judge evaluates the database the analyst used, whether the product rule was applied correctly, and whether the resulting number is presented in proper context. Testimony that fails on any reliability factor is excluded before the jury sees it. That pretrial filter is the legal system’s main defense against the kinds of errors that drove Collins.
Benford’s Law in Fraud and Tax Cases
Forensic accountants use a mathematical principle called Benford’s Law to spot fabricated financial records. In most naturally occurring datasets, the leading digit is far more likely to be small than large. The digit one appears first about 30 percent of the time; the digit nine appears first less than five percent of the time. The distribution holds across tax returns, expense reports, population figures, and many other real-world datasets.
When people invent numbers, they tend to spread leading digits more evenly than nature does, or they cluster entries just below a reporting threshold. An expense ledger showing an unusual spike in entries starting with seven or eight raises the possibility that someone manufactured transactions. The math alone does not prove fraud. It tells auditors where to dig.
Tax authorities use similar analysis during audits. Federal tax evasion is a felony carrying up to five years in prison and fines up to $100,000 for individuals, and Benford’s Law analysis of reported income and deductions can provide the initial red flag that triggers subpoenas for bank records and a full investigation. In securities fraud litigation, expert witnesses present Benford’s Law deviations to show patterns of misreported earnings. A company whose quarterly revenue figures consistently violate the expected digit distribution has some explaining to do, and the deviation gives plaintiffs a statistical foundation for focusing discovery on the periods and accounts where the numbers look least natural.
Measuring Gerrymandering With Math
When a political party draws electoral maps to lock in an advantage, courts need objective tools to measure the manipulation. Several mathematical metrics have become standard evidence in gerrymandering cases, each capturing a different dimension of fairness.
Compactness Scores
The Polsby-Popper test is the most widely used compactness measure. It compares a district’s area to the area of a circle with the same perimeter, producing a score between zero and one. A score close to one means the district is compact and roughly circular. A score near zero signals an irregular, sprawling shape, often a sign that borders were drawn to pull in favorable voters while excluding unfavorable ones. Courts treat unusually low compactness scores as circumstantial evidence of gerrymandering, though no single score automatically invalidates a map.
The Efficiency Gap and Mean-Median Difference
The efficiency gap measures how efficiently each party converts votes into seats. It counts “wasted” votes for both sides, meaning votes cast for losing candidates and votes for winners beyond what was needed to win, and calculates the difference as a percentage. When one party’s wasted-vote share is consistently much larger, the map likely favors the other party.
The mean-median difference is a simpler diagnostic. It compares a party’s average vote share across all districts with its vote share in the median district. A significant gap between the two means the distribution of voters is skewed, so one party must win a larger share of the statewide vote just to capture half the seats. Both metrics have been accepted by federal and state courts as useful tools in gerrymandering challenges.
The Limits After Rucho
In Gill v. Whitford (2018), the Supreme Court heard a challenge to Wisconsin’s state legislative map built heavily on the efficiency gap but decided the case on standing without reaching the merits. A year later, in Rucho v. Common Cause (2019), the Court closed the federal courthouse door on partisan gerrymandering claims altogether, holding that these disputes present political questions beyond the reach of federal courts and that no manageable judicial standard exists for deciding when partisan advantage crosses a constitutional line. Mathematical metrics like the efficiency gap remain powerful analytical tools, but federal judges can no longer use them to strike down maps on partisan-gerrymandering grounds. Challenges based on racial gerrymandering remain viable, and state courts applying their own constitutions can still rely on these metrics.
Risk-Assessment Algorithms at Sentencing
Mathematical models now influence bail, sentencing, and parole decisions across the country. Tools like COMPAS, which stands for Correctional Offender Management Profiling for Alternative Sanctions, assign defendants a numerical risk score based on factors such as age, criminal history, and survey responses. Judges then use those scores alongside other information when deciding how to sentence someone or whether to grant pretrial release.
The Wisconsin Supreme Court addressed these tools in State v. Loomis (2016), ruling that judges may consider a COMPAS risk score at sentencing as long as they observe specific limitations. Risk scores cannot serve as the deciding factor in whether someone goes to prison or how long the sentence runs. Every sentencing court must identify factors beyond the risk score that independently support the sentence. Presentence reports containing a COMPAS assessment must include a written warning about the tool’s limitations.
Those limitations are substantial. COMPAS is proprietary, so defendants cannot fully examine how their score was calculated. The tool was designed for corrections departments making treatment and supervision decisions, not for courtroom sentencing. Its scores reflect group-level data, identifying categories of higher-risk people rather than predicting any individual’s behavior. Researchers have also documented cases where defendants with long criminal histories received low scores and vice versa, raising basic accuracy concerns. Independent analyses have questioned whether the tool disproportionately classifies minority defendants as high risk, though the extent and cause of any racial disparity remains actively debated. Algorithmic risk tools promise objectivity and deliver opacity, and because their formulas are protected as trade secrets, defendants often have no practical way to challenge the math behind their own score. The constitutional boundaries around algorithmic sentencing remain unsettled.
Present-Value Math for Future Damages
When someone is seriously injured and will lose income for years or need ongoing medical care, courts translate a stream of future losses into a single lump sum. The calculation runs in two layers: how long the losses will last, and what those future losses are worth today.
Courts set the time horizon using actuarial life tables published by the Centers for Disease Control and Prevention and the Social Security Administration. The tables give statistical life expectancy based on age and sex, and both sides may present adjusted figures accounting for the injured person’s health before the accident, preexisting conditions, and lifestyle. That number drives the calculation of future medical expenses, lost earning capacity, and long-term care.
Reducing those future losses to present value recognizes that a dollar today is worth more than a dollar ten years from now. The calculation balances a growth rate, meaning expected increases in wages and medical costs, against a discount rate, meaning the return the plaintiff could earn by investing the lump sum. When the two rates are assumed equal, under the “total offset” approach, the math simplifies to the annual loss multiplied by the number of years. When they differ, economists build more complex models using Bureau of Labor Statistics compensation data and market interest rates. Some jurisdictions mandate the total-offset approach; others leave the methodology to the parties’ competing experts. Economists typically add roughly 27 to 30 percent on top of lost wages for fringe benefits such as employer-provided health insurance and retirement contributions. A discount rate off by a fraction of a percentage point compounds dramatically over a 30-year projection, which is why these calculations are often the most contested piece of math in a personal injury trial.
The Line Law Cannot Cross
In 1897, the Indiana state legislature nearly tried to redefine mathematics itself. House Bill 246, now remembered as the Indiana Pi Bill, was the work of Edward J. Goodwin, a physician who believed he had found a method for “squaring the circle,” a geometric construction mathematicians had already proved impossible. Among other errors, Goodwin’s bill implied that pi equaled 3.2 rather than its actual value of roughly 3.14159. The bill passed the Indiana House unanimously. It reached the Senate while Clarence Abiathar Waldo, a mathematics professor at Purdue University, happened to be visiting the statehouse on unrelated business. Waldo explained the impossibility of Goodwin’s claims to the senators, who quietly tabled the bill. It was never voted on and never became law.
The episode captures the principle running through everything above. Courts can decide how mathematical evidence is used, who may present it, and what standards it must meet. They cannot change the underlying numbers. The cases that go wrong are almost always the ones where someone, whether a prosecutor, an algorithm designer, or a state legislator, forgets that distinction.