The Concept of Random Match Probability

Random match probability answers a specific question: if you picked one person at random from the relevant population, what is the probability that person would match the evidence? The question is not about the defendant. It is a statement about the rarity of the evidence characteristic in the population, nothing more.

The question RMP does not answer is: given that we observe a match, what is the probability the defendant is the source? That posterior probability depends on three things: the RMP, the size and composition of the realistic alternative suspect pool, and any other evidence in the case. None of those three inputs can be derived from the RMP alone. A forensic scientist who reports an RMP without acknowledging this distinction is giving the fact-finder an incomplete picture.

The population underlying the RMP also requires definition. A DNA STR profile RMP calculated from a South Asian reference database will differ from one calculated from a European database, sometimes by an order of magnitude, because allele frequencies differ across populations. The choice of reference population should reflect the realistic pool of potential contributors, not the most favourable or least favourable database for the prosecution. In practice, many laboratories report RMP from multiple reference databases and cite the least favourable (largest) estimate as the primary figure. See Population Databases for Forensic Statistics for database selection criteria.

For a standard short tandem repeat (STR) DNA profile, RMP is computed using the product rule. At each locus, the genotype frequency is estimated from allele frequencies in a reference database. For a heterozygous genotype with alleles a and b at a given locus, the genotype frequency is 2 * p(a) * p(b). For a homozygous genotype with allele a, it is p(a)^2. These per-locus frequencies are then multiplied together across all typed loci to give the profile frequency.

The product rule assumes statistical independence of allele frequencies across loci. For loci on different chromosomes (unlinked loci), this assumption is well supported. For closely linked loci on the same chromosome, linkage disequilibrium can cause the product rule to underestimate or overestimate the true profile frequency, and practitioners must use databases or methods that account for this.

The theta (FST) correction accounts for population sub-structure. Within any broadly defined population (say, South Asian or sub-Saharan African), there are sub-groups whose allele frequencies differ from the overall average. Using the raw average frequency underestimates the true genotype frequency for someone from a sub-group where that allele is more common. Applying a theta correction, with theta typically set at 0.01 to 0.03, makes the RMP larger (more conservative, more favourable to the defendant) by inflating the effective allele frequency. The National Research Council (1996) report and subsequent guidance from laboratories in the UK, US, and Australia all recommend applying theta.

The likelihood ratio is the preferred currency of forensic evidence evaluation in most contemporary laboratory reporting guidelines, including those issued by the Forensic Science Regulator in England and Wales, the Scientific Working Group for DNA Analysis Methods (SWGDAM) in the United States, and the European Network of Forensic Science Institutes (ENFSI). The LR expresses the evidential weight of a match as a ratio: how many times more probable is the evidence under the prosecution hypothesis than under the defence hypothesis?

Under the simplest scenario, a single-source profile where the crime-scene sample matches the defendant and there is no transfer or persistence issue, the prosecution hypothesis is that the defendant deposited the sample. The probability of observing the match under this hypothesis is 1 (if the defendant is the source, we expect a match). The defence hypothesis is that an unknown, unrelated person deposited the sample. The probability of observing the match under this hypothesis is the RMP. So LR = 1 / RMP. An RMP of 1 in a billion gives an LR of 1 billion: the match is a billion times more probable if the defendant is the source than if a random person is.

The LR framework also generalises naturally to evidence other than DNA. Fingerprint experts in several jurisdictions now report their conclusions as LRs rather than categorical match/no-match opinions. Toolmark, glass fragment, fibre, and shoeprint evidence can all be evaluated using the same LR structure, though the underlying probability models differ. The role of statistics in evidence evaluation across these disciplines is examined in Role of Statistics in Evidence Evaluation.

Bayes' theorem relates prior probability to posterior probability through the likelihood ratio. In the context of a forensic match: the posterior odds that the defendant is the source, given the evidence, equal the prior odds multiplied by the LR. Symbolically: posterior odds = prior odds × LR. The prior odds represent everything the fact-finder believed about the defendant's culpability before hearing the forensic evidence. The LR then updates that belief.

This framework makes explicit why RMP alone cannot establish guilt or innocence. Suppose the LR is 10 million (corresponding to an RMP of 1 in 10 million). If the prior odds were 1 to 1,000 (say, the defendant was one of 1,000 plausible suspects with no other distinguishing evidence), the posterior odds are 10 million / 1,000 = 10,000 to 1 in favour of the defendant being the source. If the prior odds were 1 to 1 (the defendant was the only realistic suspect given other strong evidence), the posterior odds become 10 million to 1. The RMP is the same in both cases; the probability of guilt is very different.

Courts in most jurisdictions do not ask forensic scientists to state the prior probability, and they should not do so: the prior is the province of the jury or judge, who must weigh it against all other evidence. The forensic scientist's job is to provide a correctly stated LR and explain what it means. The Bayesian framework is the logical structure within which the jury then operates, even if the jury is not told to think in terms of Bayes' theorem. The UK Court of Appeal addressed this in R v Adams [1996] and subsequent cases, reinforcing that experts should not usurp the fact-finding function by combining the LR with a prior to give a final probability.

The prosecutor's fallacy occurs when the conditional probability P(evidence | defendant not the source) is presented or understood as P(defendant not the source | evidence). In practice, an expert witness says something like: 'The chance of this DNA profile belonging to someone other than the defendant is one in a billion.' This implies the chance the defendant is innocent is one in a billion, which is not what the RMP says. The RMP says the profile is rare in the population; it does not account for the size of the suspect pool, the strength of other evidence, or the possibility of transfer.

The fallacy has been identified in courts across jurisdictions. In the UK, the case of R v Deen (1993) resulted in a conviction being quashed after the prosecution expert incorrectly stated that the DNA evidence meant there was only a one-in-three-million chance the defendant was not the rapist. In the US, the NRC II report (1996) explicitly addressed the fallacy and called for clear, standardised language in expert testimony. In Australia, the Frequentist Interpretation Committee of the International Association for Forensic Sciences has produced guidance on correct terminology.

The defence fallacy runs in the opposite direction. Defence counsel argues: the RMP is 1 in 10 million, but there are 1.4 billion people in India (or 330 million in the United States), so there must be hundreds or thousands of people who match, meaning the evidence proves nothing. This argument misrepresents the relevant population. The alternative suspect pool is not the entire country; it is the set of people who could realistically have deposited the sample at that time and place. In most cases that pool is very much smaller, and restricting the calculation to it typically preserves substantial evidential weight.

Evaluative reporting guidelines from several jurisdictions now require forensic scientists to express their conclusions as likelihood ratios on a named verbal scale rather than as bare RMP figures. The ENFSI (European Network of Forensic Science Institutes) verbal scale runs from 'limited support' (LR 1 to 10) through 'moderate', 'moderately strong', 'strong', 'very strong', to 'extremely strong support' (LR greater than 10 million). The Forensic Science Regulator in England and Wales endorses the ENFSI approach. The Association of Forensic Science Providers (AFSP) in the UK has issued standards requiring that expert reports state the propositions clearly, report the LR, translate it into a verbal equivalent, and not state a conclusion about guilt or innocence.

In India, forensic DNA evidence is admissible under the Bharatiya Sakshya Adhiniyam 2023 (which replaced the Indian Evidence Act 1872), with Section 79A and related provisions governing electronic records and expert opinion. The statute does not mandate a specific reporting format for probabilistic evidence, but case law from the Supreme Court of India has emphasised that DNA evidence, while powerful, must be evaluated alongside other evidence and cannot substitute for proof of identity beyond reasonable doubt on its own. Courts in the United States similarly apply the Daubert standard (or the older Frye standard in some states) to evaluate whether the probabilistic methods used to calculate RMP are sufficiently reliable and generally accepted. The Federal Rules of Evidence govern admissibility, and federal courts have repeatedly examined the adequacy of population databases and the product rule.

The cold-hit problem is a specific application of RMP that arises when a crime-scene profile is searched against a large database and a match is found. If a database of one million profiles is searched and the RMP is 1 in 100,000, the expected number of coincidental matches in the database is 10. The post-search probability that any specific hit is a genuine match is substantially lower than the pre-search RMP implies. Correct analysis requires adjusting the prior probability to reflect the database search, an adjustment that has been accepted in courts in England, the United States, and Australia but is not yet uniformly applied. See Numbers in Forensic Conclusions for how probabilistic numbers are communicated in final reports.