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The multiplication rule 81
example, what is the probability that both the New York and the London
Stock Market indices will fall today, or what is the probability that we
will suffer strikes this month at both of our two production plants? The
probability of A and B occurring is known as a joint probability,andjoint
probabilities can be calculated by using the multiplication rule.
Before applying this rule we need to establish whether or not the two
events are independent. If they are, then the multiplication rule is:
p(A andB) = p(A) × p(B)
For example, suppose that a large civil engineering company is involved
in two major projects: the construction of a bridge in South America and
of a dam in Europe. It is estimated that the probability that the bridge
construction will be completed on time is 0.8, while the probability that
the dam will be completed on time is 0.6. The teams involved with the
two projects operate totally independently, and the company wants to
determine the probability that both projects will be completed on time.
Since it seems reasonable to assume that the two completion times are
independent, we have:
p(bridge and dam completed on time) = p(bridge completed on time)
× p(dam completed on time)
= 0.8 × 0.6 = 0.48
The use of the above multiplication rule is not limited to two independent
events. For example, if we have four independent events, A, B, C
and D, then:
p(A and B and C and D) = p(A) × p(B) × p(C) × p(D)
If the events are not independent the multiplication rule is:
p(A andB) = p(A) × p(B|A)
because A’s occurrence would affect B’s probability of occurrence. Thus
we have the probability of A occurring multiplied by the probability of
B occurring, given that A has occurred.
To see how the rule can be applied, consider the following problem.
A new product is to be test marketed in Florida and it is estimated that
there is a probability of 0.7 that the test marketing will be a success.
If the test marketing is successful, it is estimated that there is a 0.85
probability that the product will be a success nationally. What is the