(2)
Assume now that the first coin is tails, on 50% of occasions the second
coin is also tails. Hence the probability of two tails is 50% of 50%, namely
25%.
Statement of Rule
Having shown how the product or multiplication rule is derived, we can now
formally state the rule. When two or more events are independent the
probability that all of the events will occur is equal to the multiple of the
probability of each event. Thus if the events (designating an event by E) and
their probabilities are E¹
(A%), E² (B%) … En (Q%), the probability of all of
them happening (E¹, E² and En) is A% x B% x … Q%.
In formal terms, the rule can be stated in the following way. If A and B are
independent events, the probability that both A and B occur simultaneously is
P(A)P(B).
Illustration of Rule
One area in which the product rule is used is the process of finding facts in a
court case. Therefore we will illustrate the rule by showing how it is used to do
this.
For this illustration assume that there is a case with the following
characteristics:
(1)
The case involves a cause of action which has four elements, Elements
1-4.
(2)
Facts which satisfy these elements are Facts 1-4. Thus Fact 1 satisfies
Element 1, Fact 2 satisfies Element 2, Fact 3 satisfies Element 3 and Fact 4
satisfies Element 4.
(3)
We assume that the facts are independent.
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(4)
We assume that this is a civil action where the standard of proof is
proof on the balance of probabilities, that is.
proof to a probability of 51%.
Although we assume that this is a civil action the reasoning applies equally to a
criminal action, although we have to adjust the standard of proof to the
criminal standard, namely proof beyond all reasonable doubt.
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___________________
238
In Hodgson (1995) pp 746-748, Justice David Hodgson says that in reality
facts are more likely to be dependent than independent. But because both are
possible “courts should generally be aware of the underlying mathematical rules”
(p 748). See also Rose v Abbey Orchard Property Investment [87] Aust Torts Reports
¶80-121.
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To avoid confusion, it is worth noting that proof “beyond reasonable”
doubt can potentially be stated as a percentage, but there is no formal designation
of what that percentage is. In principle, in the author’s view, it should be around
99%.
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