082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

910 Chapter 28 Probability
Example28.1 Using the data in Table 28.1 calculate the probability that a component selected at random
iseither standard or topquality.
Solution On average 18 out of 100 components are top quality and 65 out of 100 are standard
quality.So83outof100areeithertopqualityorstandardquality.Hencetheprobability
that a component is either top quality or standard quality is 0.83. The solution may be
expressed more formally as follows. LetAbe the event that a component is top quality.
LetBbe the event thatacomponent isstandard quality.
Then,
P(A) = 0.18 P(B) = 0.65
P(A ∪B) = 0.18 +0.65 = 0.83
Note thatinthisexample
P(A ∪B) =P(A) +P(B)
In Example 28.1 the eventsAandBcould not possibly occur together. A component is
either top quality or standard quality but cannot be both. We sayAandBare mutually
exclusivebecausetheoccurrenceofoneexcludestheoccurrenceoftheother.Theresult
applies more generally.
E
E i
E j
If the occurrence of either of eventsE i
orE j
excludes the occurrence of the other,
thenE i
andE j
aresaidtobemutuallyexclusive events.
IfE i
andE j
aremutuallyexclusive wedenotethis by
Figure28.5
E i andE j are mutually
exclusive eventsand so
are depicted asdisjoint
sets.
E i
∩E j
= ◦/
Weuse◦/todenotetheemptyset,thatisasetwithnoelements.Ineffectwearestating
that the compound eventE i
∩E j
is an impossible event and so will never occur. On a
Venn diagramE i
andE j
are shown as disjoint sets (see Figure 28.5). Suppose thatE 1
,
E 2
,...,E n
arenevents and that in a single trial only one of these events can occur. The
occurrenceofanyevent,E i
,excludestheoccurrenceofallotherevents.Sucheventsare
mutuallyexclusive.
Formutuallyexclusive events the addition lawofprobabilityapplies:
P(E 1
orE 2
or ... orE n
)=P(E 1
∪E 2
∪···∪E n
)
=P(E 1
)+P(E 2
)+···+P(E n
)