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

28.3 Mutually exclusive events: the addition law of probability 911
Engineeringapplication28.3
Electricalcomponentreliability
Electrical components fail after a certain length of time in use. Their lifespan is not
fixed but subject to variation. It is important to be able to quantify the reliability of
these components. Consider the following problem.
Thelifespansof5000electricalcomponentsaremeasuredtoassesstheirreliability.
The lifespan (L) is recorded and the results are shown in Table 28.2. Find the
probability thatarandomly selected component will last
(a) more than 3 years
(b) between 3 and 5 years
(c) lessthan 4 years
Table28.2
Thelifespansof5000electricalcomponents.
Lifespanofcomponent(years)
Number
L>5 500
4<L5 2250
3<L4 1850
L3 400
Solution
We define eventsA,B,C andD:
A: the component lasts more than 5 years
B: the component lasts between 4 and 5 years
C: the component lasts between 3 and 4 years
D: the component lasts 3 years or less
P(A) = 500 2250 1850
= 0.1, P(B) = = 0.45, P(C) =
5000 5000 5000 = 0.37,
P(D) = 400
5000 = 0.08.
The eventsA,B,C andDare clearly mutually exclusive and so the addition law
may be applied.
(a) P(component lasts more than 3 years) =P(A ∪B∪C)
=P(A) +P(B) +P(C)
=0.1 +0.45 +0.37
=0.92
There isa92% chance a component will lastformore than 3 years.
(b) P(component lasts between 3 and 5 years) =P(B ∪C)
=P(B) +P(C)
=0.45 +0.37
=0.82 ➔