Case Study 1: People v. Collins — The Birth of the Prosecutor's Fallacy in Court

A structured analysis. The facts below are drawn from the published California Supreme Court decision, a matter of public record and a Tier-1 source. People v. Collins predates forensic DNA, but it is the single most important case for understanding DNA statistics, because it is where a court first dissected, by name, the exact reasoning error that DNA evidence makes so tempting.

Why this case, in a DNA-statistics chapter?

Because the prosecutor's fallacy was born here, in full view, before DNA existed — and naming it without a DNA profile makes the logic unmistakable. Collins is the controlled experiment: strip away the genome and the capillary electrophoresis, and you can see the bare statistical error that, decades later, would attach itself to DNA testimony and help convict the innocent. Every analyst who reports a random match probability is one careless sentence away from doing what the prosecutor in Collins did. Studying it is inoculation.

Background and the evidence

In June 1964, an elderly woman was assaulted and robbed in an alley in the San Pedro area of Los Angeles. She and a witness gave descriptions: a young woman with blonde hair in a ponytail fled the scene and got into a yellow automobile driven by a Black man with a beard and mustache. A married couple matching these general descriptions — Janet and Malcolm Collins — were arrested and charged.

The physical evidence connecting this specific couple to the crime was thin. What the prosecution offered instead was a piece of mathematics. To shore up the identification, the state called a college mathematics instructor and walked the jury through a calculation built on the product rule — the same multiplication of probabilities you learned in §9.1, but applied to eyewitness characteristics rather than independent genetic loci.

🔬 At the Bench — what the prosecution did with the numbers The prosecutor assigned a probability to each described characteristic and multiplied them together. The figures used in the argument were, illustratively and as reported in the decision, along these lines: a yellow automobile (1 in 10), a man with a mustache (1 in 4), a woman with a ponytail (1 in 10), a woman with blonde hair (1 in 3), a Black man with a beard (1 in 10), an interracial couple in a car (1 in 1,000). Multiplied together, these produced a headline figure: roughly 1 in 12 million that a random couple would possess all these traits. The prosecutor invited the jury to treat that figure as the probability that the Collinses were innocent — that only one couple in 12 million could be the guilty pair, and here they were.

The couple was convicted. The case reached the California Supreme Court, which reversed — and the opinion became a landmark taught in statistics and law courses ever since.

What the evidence did not establish — the two fatal flaws

The court identified two independent, fatal problems with the prosecution's statistical argument. Both map directly onto this chapter.

Flaw 1 — The probabilities were fabricated and the traits were not independent

The individual probabilities (1 in 10 for a yellow car, and so on) had no empirical foundation. They were not measured; they were asserted. This is the Collins version of a lesson the book repeats: a frightening number assembled from made-up inputs is not science, however rigorous the multiplication looks.

Worse, the product rule requires the events being multiplied to be independent — and several of the traits plainly were not. "A man with a beard" and "a Black man with a beard" overlap heavily; "a woman with a ponytail" and "a woman with blonde hair" are not independent of each other or of the other descriptors. Multiplying non-independent probabilities double-counts and drives the product to a spuriously tiny number. This is the exact danger §9.1 warned about: the multiplication is only valid when the factors are genuinely independent — which, for properly chosen DNA loci, population geneticists have worked hard to establish, and which the prosecution in Collins simply assumed.

⚠️ Junk-Science Alert Collins is the original cautionary tale for the product rule. The multiplication that makes DNA statistics legitimately powerful (independent loci, frequencies estimated from real population data) is the same multiplication that makes the Collins argument worthless (invented frequencies, correlated traits). The arithmetic is identical; the difference is entirely in whether the inputs are real and the factors independent. When you see a tiny probability built by multiplication, your first two questions are always: Where did each factor come from? And are they actually independent?

Flaw 2 — The prosecutor's fallacy: coincidence is not guilt

Even if the 1-in-12-million figure had been sound, the court recognized a second, deeper error — the one that gives this chapter its name. The prosecution treated "the probability that a random couple matches this description" as if it were "the probability that the Collinses are innocent." Those are different quantities (§9.4). The 1-in-12-million figure, taken at face value, is a statement about how rare the description is — about coincidence. It says nothing, by itself, about whether this couple, rather than some other couple who also happened to match, committed the crime.

The court made the point with a now-famous observation: in a large enough population, even a 1-in-12-million description could be expected to fit more than one couple. If two or more couples in the area matched, the probability that the Collinses specifically were the guilty ones — given only that they matched — was far from certainty. The match made them part of a set of possible matchers; it did not single them out. To leap from "they match a rare description" to "they are almost certainly guilty" was to commit the transposed conditional — to confuse $P(\text{match} \mid \text{innocent})$ with $P(\text{innocent} \mid \text{match})$.

⚖️ In the Courtroom The structural lesson is the one §9.6 turns into a rule. The prosecution's expert (and the prosecutor) reported a number about coincidence and then attached it to a conclusion about guilt — the precise move an honest expert must never make. Had the testimony stopped at "this description is rare, occurring in roughly 1 in [X] couples," and left the inference to the jury, the statistical error would have been avoided. The fallacy lives entirely in the final, unearned sentence — and removing that sentence is the whole discipline this chapter teaches.

Outcome

The California Supreme Court reversed the conviction, holding that the mathematical testimony was both factually unfounded (the made-up, non-independent probabilities) and logically fallacious (the leap from coincidence to guilt), and that it had likely overwhelmed the jury with a false aura of precision. The decision did not hold that statistics can never be used in court — it held that statistics misused in these specific ways are worse than no statistics at all, because they lend a scientific gloss to an unsound inference.

The lesson for forensic DNA

Collins contains, in 1968 and without a single base pair, every statistical danger this chapter is about:

  1. A product-rule calculation is only as good as its inputs and its independence assumption. DNA earns the multiplication; Collins did not.
  2. A small match probability describes coincidence, not guilt. The transposed conditional — the prosecutor's fallacy — is the same error whether the trait is "blonde ponytail in a yellow car" or "this twenty-locus STR profile."
  3. The fallacy lives in the final sentence. Report the rarity; never state the probability of innocence or guilt. The expert who stops at strength-of-evidence is safe; the one who takes the last step is not.

When DNA typing arrived two decades later and brought legitimately tiny match probabilities into the courtroom, the temptation Collins identified came with it — now far more dangerous, because the underlying numbers were finally real. The discipline that protects against it is exactly the discipline the California Supreme Court articulated: keep the statement on the evidence, and leave the leap to guilt where it belongs.

Discussion questions

  1. The two flaws in Collins are independent — either alone would have been enough to reverse. State each in one sentence, and explain which of the two would still apply to a modern, properly computed DNA random match probability. (Hint: only one of them.)

  2. The product rule is valid for independent events and invalid for correlated ones. Give one pair of the Collins traits that are clearly not independent, and explain how multiplying them anyway distorts the final figure.

  3. Suppose the 1-in-12-million figure had been empirically sound. Using the reasoning in §9.4–9.5, explain why it still would not establish the probability that the Collinses were the guilty couple. What is the missing ingredient?

  4. Rewrite the prosecution's statistical testimony so that it would have been admissible and honest — saying only what the numbers (if sound) could support, and stopping before the fallacy.

  5. Collins is taught as a cautionary tale, but it is also, paradoxically, a vindication of careful forensic statistics: it shows a court correctly policing the boundary between coincidence and guilt. How does the same boundary apply to the cold-case mixture in this chapter's Case File, where the LR strongly supports Roy Keller as a contributor? What is the Collins lesson there?

  6. Compare the Collins error to the defense fallacy (§9.4). Both involve a population of potential matchers. How does the defense fallacy use that same population of matchers to reach the opposite unwarranted conclusion — and why are both wrong?