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LAW EXTENSION COMMITTEE
UNIVERSITY OF SYDNEY
2009
JURISPRUDENCE OUTLINE
ALL STUDENTS PLEASE NOTE:
The outline below is intended to assist students in following the lectures and in
understanding the recommended reading. The outline is not a substitute for the lectures
and reading. The outline is not intended to be comprehensive. Students who have merely
familiarised themselves with the outline but not attended the lectures and read the
prescribed text and readings will be inadequately prepared for the exam and at
substantial risk of failure.
Examination questions will ask students to apply the concepts and arguments taught in
the course to an issue or problem. Students will be best prepared to deal with the paper
who have attended the lectures or weekend schools and read widely.
LECTURE 9
LEGAL REASONING ABOUT FACTS AND PROBABILITY THEORY
Introduction
Judges and lawyers spend more time reasoning about factual matters then legal matters and it
is therefore important to see what light contemporary philosophy may shed on this process.
The key legal concept regarding factual inquiry is the notion of judicial proof. This is
presently represented, at least in part, by the definition of relevant evidence (s.55 Evidence
Act) and the definition of standard proof (Evidence Act, s.140 civil standard, s.141 criminal
standard, being the well known rule in civil cases that proof is on the balance of probabilities,
and in criminal cases that the prosecution must prove the case beyond reasonable doubt).
The usual definition of the civil standard and the definition of relevance both use the term
“probability”. The concept of probability has been a key focus of philosophical inquiry for
several centuries.
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May 2009
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Historical Background
The mathematical calculus of probabilities was first developed by Blaise Pascal (1623-1662).
Pascal developed the mathematics of chances as an axiomatic system in a correspondence
with Fermat. This correspondence consisted of five letters exchanged in the Summer of 1654.
Pascal’s Probability Theory is frequently taught within courses on statistics and the theory has
been most often applied in calculating the likelihood of outcomes in regard to repeated events.
(eg, dice or card games or in predicting the likelihood of certain properties being possessed by
members of a population). These applications are frequently described as the frequentist
interpretation of probability theory.
Many considered puzzling the application of probability to unique events. What does it mean
to say that the probability that the accused is guilty is 90% or the probability that the
government will win the next election is 50%? In 1954 the mathematician LJ Savage in “The
Foundations of Statistics” showed that meaning could be attached to questions about the
probability of a unique event. Savage proposed that an individual choosing between
competing theories chose as if deciding what wagers to make in a lottery in which the lottery
outcomes represent the truth or falsity of the propositions. Savage was able to use rational
choice theory to show that a unique number P(A) may be assigned to a proposition A and that
the number so assigned would follow all the usual rules of conventional probabilities. This is
sometimes referred to as the subjective interpretation of probability, since the probability
assigned to the truth of an event represents the subjective judgment as to the likelihood of the
event being true.
Law and Probability Theory
The law appears to incorporate and depend upon the concept of probability in a number of
important respects. Apart from the concepts of proof the law has frequently applied
probabilistic concepts to deal with the notion of proof in situations of risk or uncertainty, see
in particular: TNT Management v Brooks (1979) 23 ALR 395 – uncertainty about the cause of an
accident;
Rose v Abbey Orchard Property Investments [1987] Australian Torts Reports 80-121 –
uncertainty about whether a precaution would have prevented an accident; and
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May 2009
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Sellars v Adelaide Petroleum (1994) 179 CLR 332 – uncertainty about the amount of
loss caused by a misrepresentation.
Is all Legal Factual Reasoning Amenable to Mathematical Analysis?
Mathematical probability as been used by courts in dealing with clear situations of uncertainty
or risk, but do the rules that govern mathematical probabilities underlie all reasoning about
factual matters.
A major debate in the legal literature was triggered by the decision in People v Collins 68
Cal.dd.319, 438p.dd 33 66 cal.Rptr 497 (1969) (Enbanc) in which a Californian court initially
permitted statistical evidence of the likelihood of an accused being the person identified, to be
admitted as evidence of the improbability that the wrong person had been prosecuted. The
decision was overturned on appeal at which it was shown that incorrect use had been made of
statistical evidence.

The case was the subject of an important article by Finklestein and Fairley – “A Bayesian
Approach to Identification Evidence” in 83 Harvard Law Review, 498 (1970).
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The authors argued that the proper approach was the use of Bayes’ theorem to determine
the revised probability of the guilt of the accused in the light of fresh evidence.
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Their article was followed by the now famous riposte by Lawrence Tribe - “Trial by
Mathematics: Precision and Ritual in the Legal Process”, 84 Harvard Law Review, 1329
(1971).
Bayes Theorem
The theorem was discovered by Thomas Bayes (1702-1761). Bayes’ theorem is more readily
applicable to subjective applications of probability theory such as unique events.
Some have argued that all reasoning about facts should, to the extent that we are able,
conform to the principles of the mathematical calculus of probabilities (eg, Robinson and
Vignaux, “Probability – The Logic of the Law”, (1993), 13 Oxford Journal of Legal Studies,
547 at 462). On this view even though we may not as a matter of practice be able to explicitly
calculate probabilities in accordance with the mathematical calculus we should seek to ensure
that our usual rules and principles of inference are consistent with the mathematical calculus.
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May 2009
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The calculus may allow us to identify areas of faulty reasoning (eg, the alleged error of logic
in the High Court judgment in Chamberlain v The Queen (No.2) (1984) 153 CLR 536).
Criticisms of the Bayesian Approach
Some scholars have argued that it is misconceived to model legal reasoning about facts on the
mathematical calculus. The criticisms made by Lawrence Tribe were expanded and enlarged
by L Jonathon Cohen in “The Probable and the Provable” (1977) which argued that legal
reasoning is not based on the application of mathematical probabilism but is based upon a
different form of probability called by Cohen “inductive probability”. Cohen relied upon
several arguments and paradoxes to show why inference drawing in law is different from the
mathematical process.
The Gatecrasher Paradox
If proof on the balance of probabilities was equivalent to proving something to an amount
greater than .5 than plaintiffs should be entitled to win cases by adducing merely statistical
evidence (eg, if there are 1,000 people in a stadium where only 499 have paid admission,
anyone sued at random would be liable to judgment in the absence of other evidence).
Because we would not accede to such proof Cohen argued that proof in law cannot simply be
mathematical. (See also the blue and red bus company paradox).
The Argument about Conjunction
Where a legal action depends upon several independent elements, all of which must exist to
establish liability, application of the mathematical calculus would require us to multiply the
probabilities of each element. Where the probabilities are less than 1 (as will always be the
case) the effect of multiplying them will be to produce a small overall probability, unless each
had a very high initial probability. The law does not apply this rule but treats each matter as
proven once it is established to be more likely than not.
Cohen also argued that the rule of inference upon inference applied in Chamberlain v The
Queen is in fact the correct principle and provides a protection against error.
The Problem of Beyond Reasonable Doubt
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It is not clear what mathematical probability should correspond to a level of belief beyond
reasonable doubt in the guilt of the accused.
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Figures such as .9 or .95 (allowing for a 10% or 5% likelihood of error) seem too low.
Higher probabilities would appear to make the prosecution’s task too onerous.
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If the presumption of innocence be equated with a probability of zero then on any
application of Bayes’ theorem the probability of the accused’s guilt will always come to
zero.
Criticisms by David Hodgson
In his article “The Scales of Justice: Probability and Proof in Legal Fact Finding” (1995) ALJ
741 Hodgson argues that the mechanical application of the concepts of mathematical
probability will produce wrong or unjust results where it is applied to an inadequate evidential
basis. Further, the calculus of probabilities may not be of much assistance in determining
what is an appropriate evidential basis.
Conclusion – What Principles Ought to Guide Legal Reasoning?
It is clear from Cohen’s many examples that legal reasoning does not comply with the
calculus of mathematical probabilism. Some might suggest that this demonstrates the
irrationality of the law. It may be that jurists consistently underestimate the probative weight
of circumstantial evidence (as some American scholars have sought to show through
empirical testing).
That the law does not follow the mathematical calculus does not end the debate. It is telling
that where we are concerned with risk and uncertainty in many other areas of life the
mathematical calculus is relied upon in determining risk (eg, in risk analysis in major
engineering projects, aircraft safety and the like). Given that we can never be absolutely
certain of the guilt of an accused putting people in gaol is a risk taking activity, the rational
method for gauging those risks may at least presumptively be one that complies with the
method we apply in determining risks in other areas of life.
Christopher Birch
May 2009