Conditional Probability and Independence

The conditional probability of A given B is defined as:

P(A|B) = P(A and B) / P(B), provided P(B) > 0.

This formula says: take the probability that both A and B occur, then rescale it relative to the probability of B. The rescaling is the key idea. Conditioning on B means we restrict our attention to a smaller universe where B has occurred, and we ask what fraction of that universe also contains A.

A simple numerical example makes this concrete. Suppose a population database shows: 30% of individuals have Blood Group O. Among individuals who are also carriers of a particular genetic variant, 50% have Blood Group O. If we know someone carries the variant, the probability they have Blood Group O is not 0.30 but 0.50. Knowing one fact changed the probability of the other. That is conditional probability in action.

In forensic evidence evaluation, conditional probability appears in the likelihood ratio framework. The numerator of a likelihood ratio is P(evidence | prosecution hypothesis): the probability of seeing this evidence if the suspect is the source. The denominator is P(evidence | defence hypothesis): the probability of seeing this evidence if someone else is the source. Both are conditional probabilities, and computing either one correctly requires knowing what other conditions are being assumed.

Events A and B are independent if and only if P(A|B) = P(A). Knowing B has occurred changes nothing about the probability of A. An equivalent statement is P(A and B) = P(A) x P(B). These two formulations are mathematically equivalent: either one implies the other.

The error to avoid is confusing conceptual independence with statistical independence. Two variables may seem unrelated in concept but may be correlated in practice because of a shared underlying factor. In human genetics, the STR loci used in forensic DNA profiling are located on different chromosomes and are, by Mendel's law of independent assortment, biologically independent. However, statistical independence at the population level requires an additional condition: the population must not be structured into subgroups with systematically different allele frequencies. If it is structured, relatives and members of the same subgroup will show correlated profiles even at loci on different chromosomes.

Testing independence requires data. For genetic loci, forensic laboratories test two things. First, Hardy-Weinberg equilibrium at each locus: if genotype frequencies at a single locus match the expectation from allele frequencies under random mating, the within-locus genotype combinations are consistent with independence between the two alleles at that locus. Second, linkage disequilibrium between pairs of loci: a chi-squared test compares observed two-locus haplotype frequencies against frequencies expected under independence. Both tests are routinely performed on forensic population databases as part of their validation.

The general multiplication rule for any two events is:

P(A and B) = P(A) x P(B|A)

When A and B are independent, P(B|A) = P(B), so the rule simplifies to:

P(A and B) = P(A) x P(B)

This simple product form is the version most commonly cited in court. It is only valid when independence has been confirmed. When events are positively dependent, P(B|A) > P(B), so P(A and B) > P(A) x P(B). Using the simple product form in this situation gives a number smaller than the true joint probability. The evidence appears rarer than it actually is.

The magnitude of the error grows rapidly with the number of features multiplied. If two features are each mildly positively dependent, their joint probability under the product rule might be off by a factor of two. Multiply ten or fifteen features together, as in a full STR profile, and the error in the final number can become enormous. This is why the validation of independence between loci is not optional: it is the mathematical precondition for the product rule's validity.

Positive dependence means that two events reinforce each other: if one is true, the other becomes more likely. In a forensic context, positive dependence between features often arises from shared causes. Two hair characteristics that are both controlled by pigmentation genetics are positively dependent. Two behavioural features of a crime scene that both reflect a perpetrator's training are positively dependent. Two measurements that are both affected by the same measurement instrument error are positively dependent.

When positively dependent features are treated as independent and multiplied together, the product can be far smaller than the true probability. Suppose the true probability of seeing both features together is 1 in 1,000, but treating them as independent yields 1 in 10,000. Presented to a jury, the 1 in 10,000 figure is ten times more persuasive than warranted. Multiply across several such pairs, and the jury may hear a figure like 1 in a billion when the true figure is 1 in a thousand.

Negative dependence is less common in forensic work but does occur. If two features are mutually exclusive, knowing one is present makes the other impossible: they are maximally negatively dependent. Treating negatively dependent features as independent would overstate the joint probability and understate the strength of evidence. This direction of error is less discussed in the forensic literature, probably because it leads to conservative evidence statements rather than overstatements, but it is equally a departure from correct statistical practice.

The most widely cited case of a misapplied independence assumption in a criminal court is the prosecution of Sally Clark in England in 1999. Two of her infant sons had died, and a prosecution expert calculated the probability of two sudden infant deaths in one family as (1 in 8,543) squared, producing approximately 1 in 73 million. The squaring assumes the two events are independent. They are not: the genetic and environmental factors that contribute to sudden infant death syndrome are shared within a family, making two deaths in the same family positively correlated. The Royal Statistical Society wrote publicly to the Lord Chancellor highlighting the error. Clark's conviction was quashed by the Court of Appeal in 2003, after she had spent more than three years in prison.

A different kind of independence error emerged in early forensic DNA cases. In the 1980s and early 1990s, some laboratories calculated DNA match probabilities by multiplying single-locus probabilities without first testing the loci for statistical independence. Where population substructure existed, the resulting product was too small. The case of People v. Castro (New York, 1989) was the first in which a court held a full hearing on the statistical methodology behind DNA evidence. The judge excluded the quantitative statistics on the grounds that the population frequency calculations had not been performed correctly, while allowing the qualitative match conclusion to stand.

In India, the Bharatiya Sakshya Adhiniyam 2023 (which replaced the Indian Evidence Act 1872) governs the admissibility of scientific evidence, including DNA. Like its predecessors in the UK (Criminal Justice Act 2003, admissibility under relevance and reliability) and the US (Daubert v. Merrell Dow Pharmaceuticals, 1993, which requires judges to assess whether scientific methods have been tested and validated), Indian law places the burden on the party tendering evidence to establish its reliability. A forensic statistician who applies the product rule without validating independence between features is presenting an unvalidated calculation. Courts in all three jurisdictions have mechanisms to exclude or discount such evidence once the methodological flaw is identified.

Modern forensic DNA databases are validated against independence before use. The CODIS database in the United States, which uses 20 core STR loci, is accompanied by population genetic studies testing each locus for Hardy-Weinberg equilibrium and each pair of loci for linkage disequilibrium. The UK National DNA Database follows similar validation requirements set by the Forensic Science Regulator's Codes of Practice. The European Network of Forensic Science Institutes (ENFSI) publishes guidelines requiring that population databases used for match probability calculations be tested for the statistical properties that justify the product rule.

Where independence cannot be fully justified, correction factors are applied. The theta correction (also written as FST, the fixation index) adjusts match probabilities for population substructure by accounting for the increased probability that two members of the same subgroup share alleles by descent. The corrected formula for the probability of matching a two-allele genotype at a locus is:

[theta + (1-theta) x p1] x [2theta + (1-theta) x p2], where p1 and p2 are the allele frequencies and theta is the subpopulation coancestry coefficient. Typical values of theta used in forensic practice range from 0.01 to 0.03. The effect is to make match probabilities slightly less extreme, which is the conservative direction.

The independence question extends beyond DNA. Population databases for forensic statistics used for glass refractive index, fibre colour, or soil mineralogy must also be tested for correlations between features before those features are multiplied together. A database that shows a strong correlation between refractive index and chemical composition means that multiplying the frequency of a refractive index value by the frequency of a chemical composition value produces a joint probability that is too small. Feature correlation analysis is a routine part of building any multi-feature forensic comparison database.