The People of the State of California v. Collins was a 1968 jury trial in California. It made notorious forensic use of statistics and probability. Bystanders to a robbery in Los Angeles testified that the perpetrators had been a black male, with a beard and moustache, and a caucasian female with blonde hair tied in a ponytail. They had escaped in a yellow motor car.
The prosecutor called upon for testimony an instructor in mathematics from a local state college. The instructor explained the multiplication rule to the jury, but failed to give weight to independence, or the difference between conditional and unconditional probabilities. The prosecutor then suggested that the jury would be safe in estimating the following probabilities:
- Black man with beard: 1 in 10
- Man with moustache: 1 in 4
- White woman with pony tail: 1 in 10
- White woman with blonde hair: 1 in 3
- Yellow motor car: 1 in 10
- Interracial couple in car: 1 in 1000
These probabilities, when considered together, result in a 1 in 12,000,000 chance that any other couple with similar characteristics had committed the crime - according to the prosecutor, that is. The jury returned a verdict of guilty.
As seen in , upon appeal, the Supreme Court of California set aside the conviction, criticizing the statistical reasoning and disallowing the way the decision was put to the jury. In their judgment, the justices observed that mathematics:
The Collins Case
The Collins case is a classic example of the prosecutor's fallacy. The guilty verdict was reversed upon appeal to the Supreme Court of California in 1968.
... while assisting the trier of fact in the search of truth, must not cast a spell over him.
Prosecutor's Fallacy
The Collins' case is a prime example of a phenomenon known as the prosecutor's fallacy—a fallacy of statistical reasoning when used as an argument in legal proceedings. At its heart, the fallacy involves assuming that the prior probability of a random match is equal to the probability that the defendant is innocent. For example, if a perpetrator is known to have the same blood type as a defendant (and 10% of the population share that blood type), to argue solely on that basis that the probability of the defendant being guilty is 90% makes the prosecutors's fallacy (in a very simple form).
The basic fallacy results from misunderstanding conditional probability, and neglecting the prior odds of a defendant being guilty before that evidence was introduced. When a prosecutor has collected some evidence (for instance, a DNA match) and has an expert testify that the probability of finding this evidence if the accused were innocent is tiny, the fallacy occurs if it is concluded that the probability of the accused being innocent must be comparably tiny. If the DNA match is used to confirm guilt that is otherwise suspected, then it is indeed strong evidence. However, if the DNA evidence is the sole evidence against the accused, and the accused was picked out of a large database of DNA profiles, then the odds of the match being made at random may be reduced. Therefore, it is less damaging to the defendant. The odds in this scenario do not relate to the odds of being guilty; they relate to the odds of being picked at random.