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

906 Chapter 28 Probability
TheeventE 3
isknownasacompoundeventandonlyoccurswhenbothE 1
andE 2
occur
atthe same time.Usingset notation we can write
E 3
=E 1
∩E 2
= {1,2,3,4} ∩ {2,4,6}
= {2,4}
E
SoE 3
occurswhena2ora4isrolled.HenceE 1
∩E 2
canoccurintwooutofsixequally
likely ways.Therefore,
E 1
E 2
1
3
2
4
5
6
P(E 3
)=P(E 1
∩E 2
)= 2 6 = 1 3
Figure 28.2 shows a Venn diagram forthis example.
We could alsodefine the compound eventE 4
as
Figure28.2
E 1 ∩E 2 corresponds to
the compound eventE 1
occursandE 2 occurs.
E 4
:E 1
occurs orE 2
occurs
The eventE 4
occurs when eitherE 1
occurs orE 2
occurs,or both. We can write
So,
E 4
=E 1
∪E 2
= {1,2,3,4} ∪ {2,4,6}
= {1,2,3,4,6}
P(E 4
)=P(E 1
∪E 2
)= 5 6
If eventsAandBboth occur thenthiscompound event isdenotedA ∩B.
If either eventAoreventBoccursthenthiscompound event isdenotedA ∪B.
Engineeringapplication28.1
Powersupplytoacomputer
Computersusedinthecontroloflife-criticalsystemsoftenhavetwoseparatepower
supplies.Ifonepowersupplyfailsthentheothertakesover.LetE 1
betheeventthat
powersupply1fails,andletE 2
betheeventthatpowersupply2fails.Forthepower
supplytothecomputertofailcompletelythecompoundeventE 1
andE 2
mustoccur,
thatisE 1
∩E 2
occurs.
A system such as this is said to have dual modular redundancy. One of the
difficultieswithsuchasystemisindeterminingwhichsystemisoperatingcorrectly
andwhichunitisswitchedtobeingtheactiveone.Insomesituationsitisnecessary
to have three systems operating in parallel and then using a voting system to decide
on the correct output. This way the decision is taken based on the majority of the
threesystems, thatis, when atleasttwo outofthe threeagree.