we obtain a figure which can never be higher than the lowest individual
measure. Where X is a fraction, half of X is always less than X. (This point, of
course, applies to any fraction, not just a half.) To illustrate from the example
above, the overall probability that all facts are true is 21.6% which is still
lower, indeed considerably lower, than the lowest individual figure of 60%
(which is the figure for both Fact 1 and Fact 4).
Rule 2: Each Additional Fact Lowers Probability
Each additional fact introduces an additional probability. This additional
probability lowers the value of the final result unless its value is 100%. To
illustrate, start with Fact 1 then add the remaining facts one at a time and see
what happens. We can set out the results as follows:
Facts
Probability:
Probability:
Probability:
Additional Fact
Overall
Reduction in Overall Probability
Fact 1
60%
60%
Not applicable
Fact 1-2
80%
48%
20%
Fact 1-3
75%
36%
25%
Fact 1-4
60%
21.6%
40%
Figure 9.5 Independent Facts
We will illustrate the necessary calculations using Facts 1-3:
(1)
Overall probability is given by multiplying together the probability for
the individual facts. This is (60 x 80 x 75)%, namely 36%.
(2)
Reduction. There is a reduction from 48% for Facts 1-2, to 36%
for
Facts 1-3. This is a [(48-36)/48]% reduction, namely 25%.
Explanation of a Paradox
Looking at the individual probabilities on the surface (60%, 80%, 75%, 60%)
may suggest that the plaintiff has more than satisfied the standard of proof of
51%. This, however, is not the case when the facts are independent (as is the
assumption here) because the probability that all four facts are true is only
21.6%. As Rule 1 says, the final result when these individual probabilities are
multiplied together is lower than, often considerably lower than, the lowest
individual measure.
To illustrate this further, we will construct a hypothetical case using the lowest
individual probability in the example above, namely 60%. Assume that there is
just one other fact and the probability that this fact is true is 80%. In this case
the overall probability is (60 x 80)% namely 48%, which means that the
plaintiff fails to make the necessary standard of 51%. This is an extreme case
because there are only two facts and the second had a high probability. This
shows how easy it is to get a figure such as 60% “down” when the
probabilities are multiplied together.
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