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82 Introduction to probability
probability that the product will be both a success in the test marketing
and a success nationally?
Clearly, it is to be expected that the probability of the product being a
success nationally will depend upon whether it is successful in Florida.
Applying the multiplication rule we have:
p(success in Florida and success nationally)
= p(success in Florida) × p(success nationally|success in Florida)
= 0.7 × 0.85 = 0.59
Probability trees
As you have probably gathered by now, probability calculations require
clear thinking. One device which can prove to be particularly useful
when awkward problems need to be solved is the probability tree, and
the following problem is designed to illustrate its use.
A large multinational company is concerned that some of its assets in
an Asian country may be nationalized after that country’s next election.
It is estimated that there is a 0.6 probability that the Socialist Party will
win the next election and a 0.4 probability that the Conservative Party
will win. If the Socialist Party wins then it is estimated that there is a
0.8 probability that the assets will be nationalized, while the probability
of the Conservatives nationalizing the assets is thought to be only 0.3.
The company wants to estimate the probability that their assets will be
nationalized after the election.
The probability tree for this problem is shown in Figure 4.1. Note
that the tree shows the possible events in chronological order from left
to right; we consider first which party will win the election and then
whether each party will or will not nationalize the assets. The four routes
through the tree represent the four joint events which can occur (e.g.
Socialists win and assets are not nationalized). The calculations shown
onthetreeareexplainedbelow.
We first determine the probability that the Socialists will win and
the assets will be nationalized using the multiplication rule for dependent
events:
p(Socialists win and assets nationalized)
= p(Socialists win) × p(assets nationalized|Socialists win)
= 0.6 × 0.8 = 0.48