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Our first application of Hardy–Weinberg can be found in newspapers on a regular basis and commonly dramatized on television. A terrible crime has been committed. Left at the crime scene was a biological sample that law enforcement authorities use to obtain a multilocus genotype or DNA profile. A suspect in the crime has been identified and subpoenaed to provide a tissue sample for DNA profiling. The DNA profile from the suspect and from the crime scene are identical. The DNA profile is shown in Table 2.2. Should we conclude that the suspect left the biological sample found at the crime scene?

To answer this critical question, we will employ Hardy–Weinberg to predict the expected frequency of the DNA profile or genotype. Just because two DNA profiles match, there is not necessarily strong evidence that the individual who left the evidence DNA and the suspect are the same person. It is possible that there are actually two or more people with identical DNA profiles. Hardy–Weinberg and Mendel's second law will serve as the bases for us to estimate just how frequently a given DNA profile should be observed. Then, we can determine whether two unrelated individuals sharing an identical DNA profile is a likely occurrence.

Table 2.2 An example DNA profile for three STR (“simple tandem repeat”) loci commonly used in human forensic cases. Locus names refer to the human chromosome (e.g. D3 = third chromosome) and chromosome region where the SRT locus is found. The allele states are the numbers of repeats at that locus (see Box 2.1).

Locus	D3S1358	D21S11	D18S51
Genotype	17, 18	29, 30	18, 18

To determine the expected frequency of a one‐locus genotype, we employ the Hardy–Weinberg Eq. (2.1). In doing so, we are implicitly accepting that all of the assumptions of Hardy–Weinberg are approximately met. If these assumptions were not met, then the Hardy–Weinberg equation would not provide an accurate expectation for the genotype frequencies! To determine the frequency of the three‐locus genotype in Table 2.2, we need allele frequencies for those loci, which are found in Table 2.3. Starting with the locus D3S1358, we see in Table 2.3 that the 17‐repeat allele has a frequency of 0.2118 and the 18‐repeat allele a frequency of 0.1626. Then, using Hardy–Weinberg, the 17, 18 genotype has an expected frequency of 2(0.2118)(0.1626) = 0.0689 or 6.89%. For the two other loci in the DNA profile of Table 2.2, we carry out the same steps.

D21S11	29‐Repeat allele frequency = 0.1811
	30‐Repeat allele frequency = 0.2321
	Genotype frequency = 2(0.1811)(0.2321) = 0.0841 or 8.41%
D18S51	18‐Repeat allele frequency = 0.0918
	Genotype frequency = (0.0918)2 = 0.0084 or 0.84%

The genotype for each locus has a relatively large chance of being observed in a population. For example, a little less than 1% of Caucasian U.S. citizens (or about 1 in 119) are expected to be homozygous for the 18‐repeat allele at locus D18S51. Therefore, a match between evidence and suspect DNA profiles homozygous for the 18 repeat at that locus would not be strong evidence that the samples came from the same individual.

Table 2.3 Allele frequencies for nine STR loci commonly used in forensic cases estimated from 196 US Caucasians sampled randomly with respect to geographic location. The allele states are the numbers of repeats at that locus (see Box 2.1). Allele frequencies (Freq) are as reported in Budowle et al. (2001). Table 1 from FBI sample population.

D3S1358	vWA	D21S11	D18S51	D13S317
Allele	Freq	Allele	Freq	Allele	Freq	Allele	Freq	Allele	Freq
12	0.0000	13	0.0051	27	0.0459	<11	0.0128	8	0.0995
13	0.0025	14	0.1020	28	0.1658	11	0.0128	9	0.0765
14	0.1404	15	0.1122	29	0.1811	12	0.1276	10	0.0510
15	0.2463	16	0.2015	30	0.2321	13	0.1224	11	0.3189
16	0.2315	17	0.2628	30.2	0.0383	14	0.1735	12	0.3087
17	0.2118	18	0.2219	31	0.0714	15	0.1276	13	0.1097
18	0.1626	19	0.0842	31.2	0.0995	16	0.1071	14	0.0357
19	0.0049	20	0.0102	32	0.0153	17	0.1556
				32.2	0.1122	18	0.0918
				33.2	0.0306	19	0.0357
				35.2	0.0026	20	0.0255
						21	0.0051
						22	0.0026
FGA	D8S1179	D5S818	D7S820
Allele	freq	Allele	freq	Allele	freq	Allele	Freq
18	0.0306	<9	0.0179	9	0.0308	6	0.0025
19	0.0561	9	0.1020	10	0.0487	7	0.0172
20	0.1454	10	0.1020	11	0.4103	8	0.1626
20.2	0.0026	11	0.0587	12	0.3538	9	0.1478
21	0.1735	12	0.1454	13	0.1462	10	0.2906
22	0.1888	13	0.3393	14	0.0077	11	0.2020
22.2	0.0102	14	0.2015	15	0.0026	12	0.1404
23	0.1582	15	0.1097			13	0.0296
24	0.1378	16	0.0128			14	0.0074
25	0.0689	17	0.0026
26	0.0179
27	0.0102

Fortunately, we can combine the information from all three loci. To do this, we use the product rule, which states that the probability of observing multiple independent events is just the product of each individual event. We already used the product rule in the last section to calculate the expected frequency of each genotype under Hardy–Weinberg by treating each allele as an independent probability. Now, we just extend the product rule to cover multiple genotypes, under the assumption that each of the loci is independent by Mendel's second law (the assumption is justified here since each of the loci is on a separate chromosome). The expected frequency of the three‐locus genotype (sometimes called the probability of identity) is then 0.0689 × 0.0841 × 0.0084 = 0.000049 or 0.0049%. Another way to express this probability is as an odds ratio, or the reciprocal of the probability (an approximation that holds when the probability is very small). Here, the odds ratio is 1/0.000049 = 20 408, meaning that we would expect to observe the three locus DNA profile once in 20 408 Caucasian Americans.

Product rule: The probability of two (or more) independent events occurring simultaneously is the product of their individual probabilities.

Now, we can return to the question of whether two unrelated individuals are likely to share an identical three‐locus DNA profile by chance. One out of every 20 408 Caucasian Americans is expected to have the genotype in Table 2.2. Although the three‐locus DNA profile is considerably less frequent than a genotype for a single locus, it still does not approach a unique, individual identifier. Therefore, there is a finite chance that a suspect will match an evidence DNA profile by chance alone. Such DNA profile matches, or “inclusions,” require additional evidence to ascertain guilt or innocence. In fact, the term prosecutor's fallacy was coined to describe failure to recognize the difference between a DNA match and guilt (for example, a person can be present at a location and not involved in a crime). Only when DNA profiles do not match, called an “exclusion,” can a suspect be unambiguously and absolutely ruled out as the source of a biological sample at a crime scene.

Current forensic DNA profiles use 10–13 loci to estimate expected genotype frequencies. Problem 2.1 gives a 10‐locus genotype for the same individual in Table 2.2, allowing you to calculate the odds ratio for a realistic example. In Chapter 4, we will reconsider the expected frequency of a DNA profile with the added complication of allele frequency differentiation among human racial groups.