Operation of Theorem
This example above gives something of the flavour of Bayes’ Theorem. It
permits mathematical calculation of the likelihood of one event given another
event. This is how Bayes’ Theorem enters fact finding. It is relevant when
facts are based only on indirect or circumstantial evidence. For example, given
that John is a bank robber, has been in the area of a bank robbery and owns a
pistol looking like the one that the masked robber used to threaten the bank
teller, what is the probability that John was the robber on this occasion?
Bayes’ Theorem allows this probability to be calculated. However, while the
mathematical calculation involved are precise, they are based on subjective
probabilities fed into the relevant equations. Results from the equations are
flawed in that the data which they use is flawed, being mere estimates.
Nevertheless, there is a benefit in that the equations show how the specific
probabilities can be combined to give the overall probability of guilt or
innocence.
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Expected Value
A bird in the hand is worth two in the bush.
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Introduction
To explain how expected value works, assume that a firm is considering an
expansion of its market. It has two options and manages to calculate that the
potential profit for Option 1 is $250,000 and for Option 2 is $400,000.
However, there is no
certainty that this profit will eventuate. In fact, on the
information that is available to the firm, there is only a 75% chance of the
$250,000 and a 40% chance of the $400,000.
What this firm now needs is a mechanism for taking this uncertainty into
account as it faces a choice to expand or not expand its market. Indeed such a
mechanism is needed in a range of legal and non legal activities which include
whether to pass one statute or another, to litigate or not litigate, to invest in
shares or bonds, or to stay where we are rather than take a new job.
Fortunately there is such a mechanism, which is known as expected value. It
enables us to adjust the return for each possibility by factoring in the
uncertainty. We do this by measuring the return as a probable or expected
return rather than by reference simply to the dollar value of the return.
Expected value of an outcome is the probability of the outcome multiplied by
the net value of the outcome.
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Computer programs, called Bayesian Belief Networks (BBN), have been
designed to handle the complex interaction of the probabilities of multiple pieces
of evidence. For an illustration see Huygen (2002).
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This is a proverb
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