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The addition rule 77
If the events are mutually exclusive then the addition rule is:
p(A orB) = p(A) + p(B) (where A and B are the events)
For example, suppose that a manager estimates the following probabilities
for the time that a new product will take to launch:
Time to launch product Probability
1year 0.1
2years 0.3
3years 0.4
4years 0.2
Suppose that we want to use this information to determine the probability
that the launch will take either 1 or 2 years. Clearly, both events are
mutually exclusive so:
p(launch takes 1 or 2 years) = p(takes 1 year) + p(takes 2 years)
= 0.1 + 0.3 = 0.4
Similarly, if we want to determine the probability that the launch will
take at least 2 years we have:
p(launch takes 2 or 3 or 4 years) = 0.3 + 0.4 + 0.2 = 0.9
Note that the complete set of probabilities given by the manager sum to
1, which implies that the list of possible launch times is exhaustive. In
the manager’s view it is certain that the launch will take 1, 2, 3 or 4 years.
Let us now see what happens if the addition rule for mutually exclusive
events is wrongly applied. Consider Table 4.1. This relates to a tidal river
which is liable to cause flooding during the month of April. The table
gives details of rainfall during April for the past 20 years and also
Table 4.1 – The frequency of flooding of a tidal river in April over
the last 20 years
Rainfall
Light Heavy
(number of years)
Total
River flooded 4 9 13
River did not flood 5 2 7
Total 9 11 20