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

28.7 Independent events 929
E 1
,E 2
andE 3
areindependent events.
(a) P(E 1
∩E 2
∩E 3
) =P(E 1
)P(E 2
)P(E 3
) = (0.9)(0.93)(0.81) = 0.68
(b) LetE 4
andE 5
be the events
E 4
: atleastone resistor isacceptable
E 5
: no resistorisacceptable
E 4
andE 5
arecomplementary events and so
P(E 4
)+P(E 5
)=1
E 5
may be expressed as
E 5
: resistor Aisnotacceptable and resistor Bisnot acceptable and
resistor Cisnotacceptable
thatis,E 1
∩E 2
∩E 3
P(E 5
) =P(E 1
∩E 2
∩E 3
)=P(E 1
)P(E 2
)P(E 3
)
= (1 −0.9)(1 −0.93)(1 −0.81)
= (0.1)(0.07)(0.19)
= 0.00133
P(E 4
) = 1 −P(E 5
) = 1 −0.00133 = 0.99867
EXERCISES28.7
1 AandBare two independenteventswithP(A) = 0.7
andP(B) = 0.4. The compound eventAoccurs, then
Aoccurs, thenBoccurs, is denotedAAB, andother
compound eventsare denotedin a similarway.
Calculatethe probability ofthe following compound
events:
(a) AAB
(b) BAB
(c) AAAA
2 Theprobabilityacomponentis faultyis 0.07.Two
components are picked at random.Calculate the
probabilitythat
(a) both are faulty
(b) both are not faulty
(c) the firstonepicked is faultyandthe second one
picked isnot faulty
(d) atleast oneisnot faulty
3 Theprobabilityacomponentis acceptable is 0.92.
Three components are picked atrandom. Calculate
the probability that
(a) allthree are acceptable
(b) none are acceptable
(c) exactlytwo are acceptable
(d) atleast two are acceptable
4 Componentsare madebymachines A, BandC.
Machine A makes30%ofthe components,machine
B makes25% ofthe components andmachine C
makes the rest. Two components are picked at
random. Calculate the probability that
(a) they are both madebymachine C
(b) oneismadeby machine Aandoneismadeby
machine B
(c) exactlyoneofthe components ismadeby
machine B
(d) atleast oneofthe components ismadeby
machine B
(e) both components are madebythe same machine
5 Capacitorsaremanufacturedbyfourmachines,1,2,3
and4.Theprobability a capacitor is manufactured
acceptablyvaries according to the machine.The