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

28.3 Mutually exclusive events: the addition law of probability 909
acceptable components is90%. Find the probability
that acomponent selected atrandom is
(a) unacceptable
(b) acceptable and ismade bymachine A
(c) unacceptable and ismade by machine B
4 MachinesA, Band Cmake components. Machine A
makes20% ofthe components, machine B makes
30% ofthe components and machine C makesthe
rest.Theprobabilitythatacomponentisfaultyis0.07
when madeby machine A, 0.06 when made by
machine B and0.05 when made bymachine C.A
component is picked atrandom. Calculate the
probability that the component is
(a) made bymachine C
(b) made bymachine A andis faulty
(c) made bymachine B and isnotfaulty
(d) made bymachine C and isfaulty
(e) made bymachine A andis notfaulty
(f) faultyand isnotmade by machine B
5 Circuit boards are made bymachines A, B,CandD.
Machine A makes15%ofthe components, machine
B makes30%, machine C makes35%and machine D
makes the remainder. Theprobability that aboard is
acceptable is0.93 when made bymachine A, 0.96
when made by machine B,0.95 when made by
machine Cand 0.93 when made by machine D. A
board ispicked atrandom. Calculate the probability
that itis
(a) made bymachine D
(b) made bymachine A andis acceptable
(c) made bymachine B and isunacceptable
(d) made bymachine C and isacceptable
(e) made bymachine D andis unacceptable
(f) unacceptable and made bymachine C
(g) Inabatch of1000 boards,how many would be
expected to be acceptable andmade by
machine D?
Solutions
1
1 (a)
2 , 1 2 , 1 6 , 1 6 , 1 2
1
(d)
6
2 0.2, 0.2, 0.6
(b) 0
(e) 0
(c)
1
6
3 (a) 0.1 (b) 0.63 (c) 0.03
4 (a) 0.5 (b) 0.014 (c) 0.282
(d) 0.025 (e) 0.186 (f) 0.039
5 (a) 0.2 (b) 0.1395 (c) 0.012
(d) 0.3325 (e) 0.014 (f) 0.0175
(g) 186
Table28.1
Theprobability ofacar
componentfallinginto
oneoffourcategories.
Category
28.3 MUTUALLYEXCLUSIVEEVENTS:THEADDITION
LAWOFPROBABILITY
Probability
topquality 0.18
standard 0.65
substandard 0.12
reject 0.05
Consideramachinewhichmanufacturescarcomponents.Supposeeachcomponentfalls
into one offourcategories:
topquality
standard
substandard
reject
After many samples have been taken and tested, it is found that under certain specific
conditions the probability that a component falls into a category is as shown in Table
28.1. The four categories cover all possibilities and so the probabilities must sum
to 1. If 100 samples are taken, then on average 18 will be top quality, 65 of standard
quality, 12 substandardand 5 will berejected.