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

28.7 Independent events 925
5 Componentsare made by machines A, B,Cand D.
Machine Amakes17% ofthe components, machine
Bmakes 21% ofthe components, machine Cmakes
20% ofthe components and machine D makesthe
remainder. Formachine A, 96% ofthe components
are reliable, formachine B,89% are reliable, for
machine C,92% are reliable and formachine D, 97%
are reliable. Acomponent ispicked atrandom.
Calculate the probability that itis
(a) reliable
(b) notreliable
(c) reliable, given itis madeby machine B
(d) notreliable, given itis made bymachine D
(e) made bymachine Agiven itis reliable
(f) made bymachine Cgiven itis unreliable
Solutions
1 (a) 0.72 (b) 0.044 (c) 0.2576
(d) 0.2695 (e) 0.4909
2 (a) 0.8110 (b) 0.2039
4 (a) 0.925 (b) 0.4
5 (a) 0.9415 (b) 0.0585 (c) 0.89
(d) 0.03 (e) 0.1733 (f) 0.2735
3 (a) 0.53 (b) 0.153 (c) 0.1909
28.7 INDEPENDENTEVENTS
Twoeventsareindependentiftheoccurrenceofeithereventdoesnotinfluencethe
probability of the other event occurring.
Engineeringapplication28.14
Acceptabilityofmanufacturedelectronicchips
Machine 1 manufactures an electronic chip, A, of which 90% are acceptable. Machine2manufacturesanelectronicchip,B,ofwhich83%areacceptable.Twochips
are picked at random, one of each kind. Find the probability that they are both
acceptable.
Solution
The eventsE 1
andE 2
aredefined:
E 1
: chip Aisacceptable
E 2
: chip Bis acceptable
P(E 1
) = 0.9 P(E 2
) = 0.83
A single trial consists of choosing two chips at random. We require the probability
that the compound event,E 1
∩E 2
, istrue.Usingthe multiplication lawwe have
P(E 1
∩E 2
) =P(E 1
)P(E 2
|E 1
) = 0.9P(E 2
|E 1
)
➔