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

920 Chapter 28 Probability
500 = 460 will be of an acceptable standard. Of the half made by machine N, 80% ×
500 = 400 will be acceptable. Hence 860 of the 1000 components will be acceptable
andsoP(E) = 860
1000 = 0.86.Clearly,therearedistinctprobabilitiesofthesameevent;
the probability changes as the conditions change. This is intuitive and leads to the idea
of conditional probability.
We introduce a notation forconditional probability. Define eventsAandBby
A: the component ismanufactured by machine M
B: the component ismanufactured by machine N
Thentheprobabilitythatacomponentisofanacceptablestandard,givenitismanufactured
by machine M, is written asP(E|A). We read this as the conditional probability
ofE givenA. SimilarlyP(E|B) is the probability ofE happening, givenBhas already
happened.
P(E|A) = 0.92 P(E|B) = 0.8
To be pedantic, all probabilities are conditional since the conditions surrounding any
event can change. However, for many situations there is tacitly assumed a definite set
of conditions which is always satisfied. The probability of an event calculated under
onlytheseconditionsisknownastheunconditionalprobability.Iffurtherwell-defined
conditions areattached, the probability isconditional.
Engineeringapplication28.12
Productionlineproductfedbytwomachines
MachinesMandNmanufactureacomponent.Theprobabilitythatthecomponentis
of an acceptable standard is 0.95 when manufactured by machine M and 0.83 when
manufactured by machine N. Machine M supplies 65% of components; machine N
supplies 35%. Acomponent ispicked atrandom.
(a) What isthe probability thatthe component isofanacceptable standard?
(b) Whatistheprobabilitythatacomponentisofanacceptablestandardandismade
by machineM?
(c) What is the probability that the component is of an acceptable standard given it
is made by machineM?
(d) What isthe probability thatthe component wasmade bymachine M?
(e) What is the probability that the component was made by machine M given it is
ofanacceptable standard?
(f) The component is not of an acceptable standard. What is the probability that it
wasmade by machineN?
Solution
We definethe events:
A: the component ismanufactured by machineM
B: the component ismanufactured by machineN
C: the component isofanacceptable standard