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.
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.
| D3S1358 | 17, 18 |
| vWA | 17, 17 |
| FGA | 24, 25 |
| Amelogenin | X, Y |
| D8S1179 | 13, 14 |
| D21S11 | 29, 30 |
| D18S51 | 18, 18 |
| D5S818 | 12, 13 |
| D13S317 | 9, 12 |
| D7S820 | 11, 12 |
What does the amelogenin locus tell us and how did you assign an expected frequency to the observed genotype? Is it likely that two unrelated individuals would share this 10‐locus genotype by chance? For this genotype, would a match between a crime scene sample and a suspect be convincing evidence that the person was present at the crime scene?
Testing Hardy–Weinberg expected genotype frequencies
A
