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286 Methods for eliciting probabilities
techniques are likely to lead to different probability forecasts when these
are converted to a common metric.
One useful coherence check is to elicit from the decision maker not
only the probability that an event will occur but also the probability that
it will not occur. The two probabilities should, of course, sum to one.
Another variant of this technique is to decompose the probability of the
event not occurring into the occurrence of other possible events. If the
events are seen by the probability assessor as mutually exclusive then
the addition rule (Chapter 4) can be applied to evaluate the coherence
of the assessments. Such checks are practically useful and are reinforced
by the results of laboratory-based empirical studies of subjective
probability assessment, where subjective probabilities attached to sets
of mutually exclusive and exhaustive events have often been shown to
sum to less than or more than one. For example, in a probability revision
task, involving the updating of opinion in the light of new information,
one set of researchers found four out of five subjects assessed probabilities
that were greater than unity. 10 These four subjects increased their
probability estimates for likely hypotheses but failed to decrease probabilities
attached to unlikely hypotheses. Another probability revision
study found that 49 out of 62 subjects gave probability estimates for
complementary events that summed to more than unity. 11 Conversely,
another investigator asked subjects to estimate sampling distributions
from binomial populations on the basis of small samples, and found that
in most cases subjective probabilities summed to less than unity. 12
In a study addressed directly to the descriptive relevance of the
additivity axiom, Wright and Whalley 13 found that most untrained probability
assessors followed the additivity axiom in simple two-outcome
assessments, involving the probabilities of an event happening and not
happening. However, as the number of mutually exclusive and exhaustive
events in a set was increased, more subjects, and to a greater extent,
became supra-additive in that their assessed probabilities tended to add
to more than one. With the number of mutually exclusive and exhaustive
events in a set held constant, more subjects were supra-additive, and
supra-additive to a greater degree, in the assessment of probabilities
for an event set containing individuating information. In this study the
individuating background information was associated with the possible
success of a racehorse in a race that was about to start. It consisted simply
of a record of that horse’s previous performances. It seems intuitively
reasonable that most probabilistic predictions are based, in the main, on
one’s knowledge and not to any large extent on abstract notions such as
additivity. Individuating information about the likelihood of an event’s