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

908 Chapter 28 Probability
acceptable
558
1000
machine A
machine B
600
1000
machine A
machine B
600
400
not acceptable
acceptable
not acceptable
42
380
400
Figure28.3
Partial tree diagram forEngineering
application 28.2.
Figure28.4
Full tree diagram forEngineering
application 28.2.
Of the 600 components made by machine A, we know that 93% of these are acceptable,
thatis93% of 600 = 558.So 600 −558 = 42 are unacceptable.
Of the 400 components made by machine B, we know that 95% of these are acceptable,
thatis95% of 400 = 380. So 400 −380 = 20 are unacceptable.
We can now complete a full tree diagram for this problem, which is shown in
Figure 28.4.
Usingthetreediagramitisstraightforwardtocalculatetherequiredprobabilities.
(a) Theprobabilityacomponent ismadebymachineAisfoundbynotingthat600
of the 1000 components aremade by machine A,that is
P(component is made by machine A) = 600
1000 = 0.6
(b) TheprobabilityacomponentismadebymachineAandisacceptableisfoundby
notingthat558ofthe1000componentsaremadebymachineAandacceptable,
that is
P(component is made by machine Aand acceptable) = 558
1000 = 0.558
(c) P(component is made by machine Band acceptable) = 380
1000 = 0.38
(d) P(component is made by machine Band unacceptable) = 20
1000 = 0.02
20
EXERCISES28.2
1 Afair die isrolled. TheeventsE 1 , . . . ,E 5 are defined
asfollows:
E 1 :aneven number is obtained
E 2 :anodd numberis obtained
E 3 :ascore ofless than2isobtained
E 4 :a3is obtained
E 5 :ascore ofmore than3is obtained
Find
(a)P(E 1 ),P(E 2 ),P(E 3 ),P(E 4 ),P(E 5 )
(b)P(E 1 ∩E 3 )
(c)P(E 2 ∩E 5 )
(d)P(E 2 ∩E 3 )
(e)P(E 3 ∩E 5 )
2 Atrialcan have threeoutcomes,E 1 ,E 2 andE 3 .E 1
andE 2 are equally likely to occur.E 3 is threetimes
more likely to occur thanE 1 .FindP(E 1 ),P(E 2 ) and
P(E 3 ).
3 Acomponent ismadeby machines AandB.Machine
Amakes 70%ofthe components andmachine B
makesthe rest. Forboth machines, the proportion of