1. Fundamentals of Forensic DNA Typing Slides prepared by John M. Butler June 2009 Appendix 3 Probability and Statistics

2. Probability Probability is the number of times an event happens divided by the number of opportunities for it to happen (i.e., the number of trials). The concepts of probabilities can be difficult to grasp because we are often in the mind-set of thinking simply that something either happened or it did not. Probability is usually viewed on a continuum between zero and one.

3. Conditional Probability The probability that an event can occur is given by the notation or formula: P(H|E) or Pr(H|E) … This notation is shorthand for stating ‘ the probability of event H occurring given evidence E is equal to … ’ Every probability is conditional on knowing something or on something else occurring.

4. Three Laws of Probability Probabilities can take place in the range zero to one. Events that are certain have a probability of one, whereas those that are not possible have a probability of zero. If two events are mutually exclusive and we wish to know the probability that one or other of them is true then we can simply add their probabilities (the sum rule). When two events are independent of one another their probabilities can be multiplied with one another (the product rule).

5. The Likelihood Ratio

6. Statistics Statistics is a mathematical science involving the collection, analysis, and interpretation of numerical data. It provides a sense of how reliable a measurement is when the measurement is made multiple times. Statistics involves using samples to make inferences about populations.

7. Population vs. Sample A population is considered in this context to be a set of objects of interest, which may be infinite or otherwise unmeasurable in their entirety. An observable subset of a population can be referred to as a sample with a statistic being some observable property of the sample. In the context of DNA testing, the ‘population’ would be the entire group of individuals who could be considered (e.g., billions of people around the world or those living within a particular country or region). The ‘sample’ would be a set of individuals from the population at large (e.g., 100 males) who were selected at random and tested at particular genetic markers to try to establish a reliable representation of the entire population.

8. The ‘statistic’ examined might be the observed allele or genotype frequencies for the tested genetic markers.

9. Steps in Hypothesis Testing (1)formulate two competing hypotheses; (2)select the appropriate statistical model (theorem) that identifies the test statistic; (3)specify the level of significance, which is a measure of risk; (4)collect a sample of data and compute an estimate of the test statistic; (5)define the region of rejection for the test statistic; and (6)select the appropriate hypothesis.

10. Null Hypothesis vs. Alternative Hypothesis The first hypothesis is called the null hypothesis, and is denoted by H 0. The null hypothesis is formulated as an equality and indicates that a difference does not exist. The second hypothesis is usually referred to as the alternative hypothesis and is denoted by H 1 or H A. The null and alternative hypotheses are set up to represent mutually exclusive conditions so that when a statistical analysis of the sampled data suggests that the null hypothesis should be rejected, the alternative hypothesis must be accepted. Thus, the data collected (evidence gathered) should tip the scales toward either the null hypothesis or the alternative hypothesis.

11. Set up two hypotheses (H 0 and H 1 ) Collect data and calculate the test statistic (S) Look up the critical value (C) and define the region of rejection for the test statistic Is S ≤ C? yes Accept H 0 (Reject H 1 ) Accept H 1 (Reject H 0 ) no Select appropriate statistical model Specify the level of significance and its critical value (C) John M. Butler (2009) Fundamentals of Forensic DNA Typing, Figure A3.1 Hypothesis Testing

12. Truth about the population Decision based on sample examined Correct decision Type II error Type I errorCorrect decision H 0 True H 0 False Accept H 0 Reject H 0 (Accept H 1 ) Correct decision Wrongfully acquitted Wrongfully accused Correct decision (b) Example Not Guilty Guilty Courtroom Verdict Saint Sinner Defendant (a) Hypothesis Testing Decisions John M. Butler (2009) Fundamentals of Forensic DNA Typing, Figure A3.2 Decisions Based on Hypothesis Testing

13. The Meaning of Statistically Significant

14. The Bonferroni Correction