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alt="images"/>) of the consignment which is pirated. It is not practical to inspect the whole consignment so a sample of size , where is inspected.

      The frequentist method assumes that the proportion

of the consignment that is pirated is unknown but fixed. The data, that is the number of CDs in the sample that are pirated, are variable. A so‐called confidence interval is calculated. The name confidence is used since no probability can be attached to the uncertain event that the interval contains . These ideas are discussed further in Chapter 4.

The frequentist method derives its name from the relative frequency definition of probability. The probability that a particular event,

, say, occurs is defined as the relative frequency of the number of occurrences of event compared with the total number of occurrences of all possible events, over a long run of observations, conducted under identical conditions of all possible events. The limitations of such a definition are presented in Section 1.7.4.

For example, consider tossing a coin

times. It is not known if the coin is fair. The outcomes of the tosses can be used as information from which the probability of a head occurring on an individual toss may be assigned. There are two possible outcomes, heads () and tails (). Let be the number of and be the number of such that Then the probability of tossing a head on an individual toss of the coin is defined as the limit as of the fraction . The frequentist approach relies on a belief in the long‐run repetition of trials³ under identical conditions. This is an idealised situation, seldom, if ever, realised in practice. More discussion on the interpretation of such a result is given in Section 3.6.

The way in which statistics and probability may be used to evaluate evidence is the theme of this book. Care is required. Statisticians are familiar with variation, as are forensic scientists who observe it in the course of their work. Lawyers, however, prefer certainties. A defendant is found guilty or not guilty (or also, in Scotland, not proven). The scientist's role is to testify to the worth of the evidence, the role of the statistician and this book is to provide the scientist with a quantitative measure of this worth. It is shown that there are few forms of evidence that are so definite that statistical treatment is neither needed nor desirable. It is up to other people (the judge and/or the jury) to use this information as an aid to their deliberations. It is for neither the statistician nor the scientist to pass judgement (Kind 1994).

      The use of these ideas in forensic science is best introduced through the discussion of several examples. These examples will provide a constant theme throughout the book. Consideration in detail of populations from which the criminal may be thought to have come, to which reference is made in the following text, are discussed in Section 6.1.1 where they are called relevant populations. The value of evidence is measured by a statistic known as the likelihood ratio and its logarithm. These are introduced in Sections 2.3 and 2.4.

      1.3.2 Stains of Body Fluids

      Example 1.1 A crime is committed. A bloodstain is found at the scene of the crime. All innocent explanations for the presence of the stain are eliminated. A PoI is found. Their DNA profile is established and found to correspond to that of the crime stain. What is the evidential value of this correspondence? This is a very common situation yet the answer to the question provides plenty of opportunity for discussion of the theme of this book.

      Certain other questions need to be addressed before this particular one can be answered. Where was the crime committed, for example? Does it matter? Does the value of the evidence of the bloodstain change depending on where the crime was committed?

      Apart from their DNA profile, what else is known about the criminal? In particular, is there any information, such as ethnicity, which may be related to their DNA profile? What is the population from which the criminal may be thought to have come? Could they be another member of the family of the PoI?

Questions such as these and their effect on the interpretation and evaluation of evidence will be discussed in greater detail. First, consider only the evidence of the DNA profile in isolation and one particular locus, LDLR. It is no longer used in forensic genetics; it is used here for ease of explanation. Assume the crime was committed in Chicago and that there is eyewitness evidence that the criminal was a Caucasian. Information is available to the investigating officer about the genotypic distribution for the LDLR locus in Caucasians in Chicago and is given in Table 1.1. The information about the location of the crime and the ethnicity of the criminal is relevant. Genotypic population proportions vary across locations and amongst ethnic groups. A PoI is arrested and a swab is taken for a comparative genetic test. For locus LDLR the genotype of the crime stain and that of the PoI correspond. The investigating officer knows a little about probability and works out that the probability of two people chosen at random and unrelated to the suspect having corresponding alleles, using the figures in Table 1.1 as estimates of population proportions, is

      (1.1)

      (see Section 3.5.1). They are not too sure what this result means. Is it high and is a high value incriminating for the PoI? Is it low and is a low value incriminating? In fact, a low value is more incriminating than a high value.

Table 1.1 Genotypic relative frequencies for locus LDLR amongst Caucasians in Chicago based on a sample of size 200.

      Source: From Johnson and Peterson (1999). Reprinted with permissions of ASTM International.

Genotype

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