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.

To determine the expected frequency of a one-locus genotype, we employ the Hardy-Weinberg equation (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 white US 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.

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 x 0.0841 x 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 white US citizens.

Product rule The probability of two (or more) independent events occurring simultaneously is the product of their individual probabilities. Odds ratio The number of events divided by the number of non-events; one over the expected sample size required to observe a single instance or event.

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 white US citizens 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 is 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 excluded as the source of a biological sample at a crime scene.

Problem box 2.1 The expected genotype frequency for a DNA profile

Calculate the expected genotype frequency

and odds ratio for the 10-locus DNA profile

below. Allele frequencies are given in

Table 2.3.

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