1. People of California v Simpson (1995) No B A097211, Superior Court of the State of California for the County of Los Angeles; Rufo v Simpson et al, Goldman, etc. v Simpson et al, Brown, etc. v Simpson (1997) Nos SC031947, SC036340 & SC036876 respectively, Superior Court of the State of California for the County of Los Angeles.

2. R Eggleston Evidence, Proof and Probability (London: Weidenfeld and Nicholson, 2nd edn, 1983).

3. Simon, R J and Mahan, L ‘Quantifying burdens of proof (1971) 5 Law and Society Rev 319–330.CrossRefGoogle Scholar

4. Research into how real juries arrive at their decisions is prohibited in the UK, so most of the research in this field has been done in North America.

5. From Eggleston, above n 2.

6. C G G Aitken Statistics and the Evaluation of Evidence for Forensic Scientists (Chichester: Wiley, 1995).

7. Aitken asks, for example, what might the judges' 0.92 standard of proof for murder mean? If it is 0.92 to 0.08 (approximately 12:1) odds in favour of guilt, does this mean that out of every 13 Americans convicted for murder one was innocent? Aitken posits that this is unlikely to be the case, and suggests that it is more likely that ‘the judges do not have a good intuitive feel for the meaning of probability figures’.

8. Where we are interested in something happening ‘given that’ some other event has also occurred or will do in the future.

9. Pr(‘G'IG), where the vertical line signifies the given that’ condition.

10. For further discussion of such errors, see Hawkins, A S ‘Uncertain Justice’ (1993) 112(9) Law Notes 21–23.Google Scholar

11. J J Koehler ‘Probabilities in the courtroom: An evaluation of the Objections and Policies’ in D K Kagehiro and W S Laufer (eds) Handbook of Psychology and Law (New York: Springer-Verlag, 1992) pp 167–184.

12. R v Deen (1994) Times, 10 January.

13. The authors would like to make the point that they have shown the probabilities in the way in which they are typically portrayed in the literature. Thus, the correct question for the court has been stated to be ‘Given the evidence, what is the probability of the defendant being innocent?’ However, this is actually a rather worrying way of presenting the information, given that we have a system of justice which proclaims that the accused is ‘innocent until proved guilty’. We should really be interested in the probability of guilt given the evidence, ie Pr(GIEvid), not in the probability of innocence given the evidence, ie Pr(NGIEvid). Even if these two probabilities are opposite sides of the same coin, the latter formulation puts the emphasis in the wrong place.

14. People v Collins (1968) 68 Cal 2d 319, 66 Cal Rptr 497, 438 P 2d 33, 36 ALR 3rd 1176 at 34.Google Scholar

15. J Piaget and I Bärbel (trans L Leake Jr, P Burrell, H D Fischbein) La genèse de l'idée de hasard chez I ‘enfant (The origin of the idea of chance in children)’ (London: Routledge & KeganPaul, 1951).

16. E Fischbein The intuitive sources of probabilistic thinking in children (trans C A Sherrard) (Dordrecht: D Reidel Publishing Co, 1975).

17. D Kahneman, P Slovic and A Tversky (eds) Judgment under Uncertainty: Heuristics and biases (Cambridge: Cambridge University Press, 1982).

18. J M Shaughnessy ‘Research in probability and statistics: reflections and directions’ in D A Grouws (ed) Handbook on Research on Mathematics Teaching and Learning (New York: Macmillan, 1992) pp 465–494.

19. A Hawkins, F Jolliffe and L Glickman Teaching Statistical Concepts (Harlow: Longman, 1992).

20. Above n 17.

21. At the 5% level of significance.

22. The ideas of randomness and causation should not be assumed to be universal. There is certainly a cultural dimension that should not be overlooked when considering people's understanding of likelihoods, probability, etc. In some regions, for example, there is no understanding of ‘chance’, and everything is thought to have a cause, whether it be the influence of the god(s), or that of the local witch doctor, etc.

23. At the 1% level of significance.

24. Reducing the two problems to comparisons of the number of positive instances to the number of possible instances given certain prior knowledge is similar to the approach used in the tree diagrams shown in Figure 2. There, however, the comparisons were based on given (example) probability values as opposed to numbers of events. Both approaches, though, relied on a formulation of the problem that yielded an exhaustive set of possible outcomes.

25. Bayes' theorem is a formal rule for reviewing probability assessments when additional information is available. In particular, in legal contexts, it may be used to derive probabilities based on a combination of subjective beliefs and statistical evidence in the form of relative frequencies. Its importance lies in providing a link between, for example, the Pr(gui1t) and the Pr(guilt I evidence). The formal application of Bayes' theorem to item 6 follows:

(1)

Similarly,

(2)

Dividing (1) by (2) allows us to compare the probability of the cab being blue with the probability of the cab being green given the evidence of the witness who says ‘B’, ie ‘Blue’. This comparison is known as a likelihood ratio. It is not difficult to see how helpful this likelihood ratio is in determining an outcome depending on the balance of probabilities. If we replace B with Liable, G with Not Liable, and ‘B’ with Evidence in the above formulae, the resulting likelihood ratio gives a direct comparison of the relative likelihoods of liability and non-liability given the evidence.

26. Hawkins, P J and Hawkins, A S ‘Bayes Watch: Balancing the Probabilities’ (1992) 111(6) Law Notes 26–27.Google Scholar

27. Pr(B | ‘B’)/Pr(G | ‘B’)=0.8 × 0.15 / 0.2 × 0.85=0.12/0.17. This gives the ratio of the likelihoods of the two possibilities. The actual probability of the cab being Blue, given that the witness said it was Blue is Pr(B | ‘B’) = 0.12 / (0.12 + 0.17) = 0.41.

28. Bar-Hillel, M ‘The Base-Rate Fallacy in Probability Judgments’ (1980) 44 Acta Psychologica 211–233.CrossRefGoogle Scholar

29. Wagenaar, A ‘The proper seat: a Bayesian discussion of the position of expert witnesses’ (1988) 12 Law and Human Behaviour 499–510.CrossRefGoogle Scholar

30. Applying Bayes' theorem to the Grosscamp case, the required likelihood ratio of posterior probabilities is derived as:.

In fact, the judges only considered Pr(‘male’lfemale), shown in bold, confusing this with its inverse Pr(female1 ‘male’); ignoring base-rate information, Pr(male) compared to Pr(female), the prior probabilities of male and female drivers in the Netherlands; and failing to realise that a likelihoods ratio required the comparison of two probabilities, in this case Pr(malel'male‘) and Pr(femalel'male’).

31. For example, it was clear that some respondents were applying the criminal standard of proof in what was a civil case.

32. At the 0.01% level of significance.

33. See above n 11.

34. Finkelstein, M O and Fairley, W B ‘A Bayesian Approach to Identification Evidence’ (1970) 83 Harv LR 489.CrossRefGoogle Scholar

35. Tribe, L H ‘Trial by Mathematics: Precision and Ritual in the Legal Process’ (1971) 84 Harv LR 1329.CrossRefGoogle Scholar

36. Adam, R. v. [1996] 2 Cr App R 467.Google Scholar

37. R v Adurns (1997) Times, 3 November.

38. See above n 11.

39. ‘ Tehan v Us, ex re1 Shott (1966) 382 US 406 at 416.Google Scholar

40. See above n 36 at 481.

41. Above n 6.

42. Above n 26.

43. Evett, I W ‘Bayesian inference and forensic science: problems and perspectives’ (1987) 36(2) The Statistician 99–105.CrossRefGoogle Scholar

44. Lupton, S ‘On the educational value of the theory of probability’ (1892) 276 Journal of Education 357–359.Google Scholar

45. Probably the most accessible reference is P Simon, Marquis de Laplace A Philosophical Essay on Probabilities, a translation of Essai Philosophique sur les Probabilités (1812). This introduction (153 pages) to Théorie Analytique des Probabilités (645 pages) has been translated by F W Truscott and F L Emory (New York: Dover, 1951) with an introductory note by E T Bell.

46. Above n44.

47. At the 0.01% level of significance.

48. Above n 11.

49. Higher Education in the Learning Society (1997). Report of the National Committee of Inquiry, chaired by R Dearing (Full report http://lwww.leeds.ac.uk/educol/ncihe/).

50. See above n 19.

51. D Green Probability Concepts in 11–16 year old Pupils (Loughborough: Centre for Advancement of Mathematical Education in Technology, University of Technology, Loughborough, 2nd edn, 1982).