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

954 Chapter 29 Statistics and probability distributions
Engineeringapplication29.3
Samplingcomponentsmadebyamachine
Amachinemakescomponents.Theprobabilitythatacomponentisacceptableis0.9.
(a) If three components are sampled find the probability that the first is acceptable,
the second isacceptable and the thirdisnot acceptable.
(b) If three components are sampled what is the probability that exactly two are
acceptable?
Solution
Let the eventsAandBbe defined thus:
A:thecomponentis acceptable,P(A) = 0.9
B:thecomponentis notacceptable,P(B) = 0.1
(a) We denote byAAB the compound event that the first is acceptable, the second
is acceptable, the third is not acceptable. Since the three events are independent
the multiplication lawinSection 28.7 gives
P(AAB) =P(A)P(A)P(B) = (0.9)(0.9)(0.1) = (0.9) 2 (0.1) = 0.081
(b) Weareinterestedinthecompoundeventinwhichtwocomponentsareacceptable
and one is not acceptable. We denote byAAB the compound event that the first
is acceptable, the second is acceptable, the third is not acceptable. Compound
eventsABA andBAA have obvious interpretations.
If exactly two components are acceptable then either AAB or ABA or BAA
occurs.Thesecompoundeventsaremutuallyexclusiveandwecanthereforeuse
the addition law (see Section 28.3). Hence
P(exactly two acceptable components) =P(AAB) +P(ABA) +P(BAA)
From (a) P(AAB) = 0.081 and by similar reasoning P(ABA) = P(BAA) =
0.081. Hence,
P(exactly two acceptable components) = 3(0.081) = 0.243
29.10.1 Probabilityofksuccessesfromntrials
Let us now return to the general problem posed earlier. We define the compound
eventC:
C:Aoccursktimes andBoccursn −k times
( n
The k occurrences of event A can be distributed amongst the n trials in different
k)
ways (see Example 29.17). The probability of a particular distribution
(
ofkoccurrences
n
ofAandn−koccurrencesofBisp k (1−p) n−k .Sincethereare distinctdistributions
k)
possible then
P(C) =
( n
k)
p k (1−p) n−k
k=0,1,2,...,n