Page 52 - Applied Statistics And Probability For Engineers

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2-3 ADDITION RULES 35

E 1
E 2 E 4
Figure 2-11 Venn
diagram of four mutu- E 3
ally exclusive events.

Upon expanding P1A ´ B2 by Equation 2-1 and using the distributed rule for set opera-
tions to simplify P31A ´ B2 ¨ C4 , we obtain

P1A ´ B ´ C2 P1A2 P1B2 P1A ¨ B2 P1C2 P31A ¨ C2 ´ 1B ¨ C24
P1A2 P1B2 P1A ¨ B2 P1C2
3P1A ¨ C2 P1B ¨ C2 P1A ¨ B ¨ C24
P1A2 P1B2 P1C2 P1A ¨ B2 P1A ¨ C2
P1B ¨ C2 P1A ¨ B ¨ C2
We have developed a formula for the probability of the union of three events. Formulas can be
developed for the probability of the union of any number of events, although the formulas
become very complex. As a summary, for the case of three events

P1A ´ B ´ C2 P1A2 P1B2 P1C2 P1A ¨ B2
P1A ¨ C2 P1B ¨ C2 P1A ¨ B ¨ C2 (2-3)

Results for three or more events simplify considerably if the events are mutually exclu-
, E , p , E , is said to be mutually exclusive if there
sive. In general, a collection of events, E 1 2 k
is no overlap among any of them.
The Venn diagram for several mutually exclusive events is shown in Fig. 2-11. By gener-
alizing the reasoning for the union of two events, the following result can be obtained:

A collection of events, E 1 , E 2 , p , E k , is said to be mutually exclusive if for all pairs,

E i ¨ E j

For a collection of mutually exclusive events,

P1E 1 ´ E 2 ´p´ E k 2 P1E 1 2 P1E 2 2 p P1E k 2 (2-4)

EXAMPLE 2-15 A simple example of mutually exclusive events will be used quite frequently. Let X denote the
pH of a sample. Consider the event that X is greater than 6.5 but less than or equal to 7.8. This