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Population Genetics and Statistics for Forensic Analysts

Coincidence Approach

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A good description of the coincidence approach, also referred to as the random match probability, is given in Forensic DNA Evidence Interpretation,05 and reads, "The coincidence approach proceeds to offer evidence against a proposition by showing that the evidence is unlikely if this proposition is true. Hence it supports the alternative proposition. The less likely the evidence under the proposition the more support given to the alternative." The alternative is that the match occurs by chance.

The coincidence approach assesses whether or not the match between the DNA profile obtained from the evidence and the DNA profile obtained from the suspect occurs by chance (coincidence). Allele frequencies are calculated using data that assumes Hardy-Weinberg equilibrium and no linkage disequilibrium.

Read about allele frequencies in this course.

Homozygote frequencies are determined by p2 and heterozygote frequencies by 2pq (Hardy-Weinberg formulas). The STR loci used by the forensic science community are not linked, and, therefore, the genotypes from each locus can be multiplied (i.e. the product rule) to obtain the frequency of the combined individual genotypes.

Read about formulas 4.1a and 4.1b from NRC II (1996 National Research Council Report).

Watch a video on determining the genotype frequency presented by Greggory LaBerge.

As discussed previously, a theta correction can be applied to the formulas. For homozygous loci, the NRC II report recommends using Equation 4.4a with a conservative value of θ. The 2p rule was originally recommended by the NRC for VNTRs (Variable Number Tandem Repeats) based on the ambiguity of allele calls; however, since this ambiguity does not exist with STR loci it is not necessary.

The NRC II panel assumed that theta is positive for all pairs of alleles. It noted that for heterozygotes, the Hardy-Weinberg calculation is generally an overestimate. The assumption is that Hardy-Weinberg proportions always give overestimates of heterozygotes when θ > 0.

The panel surmised that for homozygotes (using equation 4.4a) with small allele frequencies, a small value of θ can introduce a large change in the genotype frequency.

Laboratories are not compelled to choose one formula over another, but analysts should be familiar with the relevant discussions on this topic.

Example Calculation

Homozygotes: p2 + p(1-p)θ (Homozygote example w/ theta)

An evidence sample has a genotype at D3S1358 of 16, 16, and a θ of 0.03 is used.

The frequency of the 16 allele in Caucasians = 0.2315.

Genotype frequency = (0.2315)2 + [(0.2315)(1-0.2315)(0.03)]

Genotype frequency = 0.0535 + [(0.2315)(0.7685)(0.03)]

Genotype frequency = 0.0535 + 0.0053

Genotype frequency = 0.0588

Heterozygote: 2papb(1- θab), a ≠ b
(Heterozygote example w/ theta)

An evidence sample has a genotype at D3S1358 of 15, 17.

The frequency of the 15 allele = 0.2904 and the frequency of the 17 allele = 0.2000 in African Americans and a θ of 0.03 is used.

Genotype frequency = [2(0.2904)(0.2000)] [(1-0.03) ]

Genotype frequency = (0.11616)(0.97)

Genotype frequency = 0.1127

Heterozygote: 2papb

An evidence sample has a genotype at D3S1358 of 15, 17.

The frequency of the 15 allele = 0.2904 and the frequency of the 17 allele = 0.2000 in African Americans.

Genotype frequency = [2(0.2904)(0.2000)]

Genotype frequency = 0.1161

The combined frequency across multiple loci (non-linked) can be calculated by using the product rule, multiplying the respective frequencies together.


Evidence sample locus with locus frequencies


Locus Frequency







  • Total frequency = (frequency of D3S1358)(frequency of vWA)(frequency of FGA)
  • Total frequency = (0.07534)(0.02725)(0.04453)
  • Total frequency = 0.0000914
  • Probability = 1/0.0000914 = 1 in 10,941 unrelated people
  • Probability of observing the given genotype is reported as 1 in 10,941 unrelated people.

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