Comment
Several important things need to be said to put this in perspective. First, the
figures are estimates (both the probability of an outcome and the likely gain or
loss). Second, this analysis of expected value can be refined. For example, the
response of a person to a risk may not be a simple comparison of expected
values but can be affected by their attitude to risk. If they are a risk taker they
will be more prepared to take risks for gain; if they are risk averse they will be
less prepared to do so, even when the expected value is positive.
Problems
In principle expected value is an extremely useful tool. However,
it has two
major limitations. First, the probability that proposed action will incur a cost or
return a benefit can rarely be known precisely. Second,
it can be difficult to
compute net benefit, because costs and benefits are not always
commensurable.
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Fallacies in Probability
Introduction
Careless thinking about probability can lead to errors. These errors involve
arguing from one established probability to a second probability. These
fallacies are of extreme concern when the second probability involves guilt or
innocence. Some common forms of this fallacious reasoning have been
identified and labelled the prosecutor’s fallacy and the defendant’s fallacy.
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Prosecutor’s Fallacy
An illustrative version of the prosecutor’s fallacy is as follows. Assume that
there is a one in two million chance of a match of evidence at the crime scene
if a defendant is innocent. Based on this, the prosecutor argues that there is a
one in two million chance of innocence and consequently a 1,999,999 in two
million (99.99995%) chance of guilt, which is well past the point of proof
beyond all reasonable doubt.
But assume further that the crime took place in a city of ten million people, any
of whom might be the perpetrator. If each person was tested there one would
expect five matches with the evidence. On this basis on the possibility of a
match taken on its own there is only a one in five (20%) chance of guilt, which
is way below reasonable doubt.
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Chapter 12 Measurement of Net Benefit
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These were identified and labelled by Thompson and Schumann (1987).
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