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

956 Chapter 29 Statistics and probability distributions
29.10.3 Mostlikelynumberofsuccesses
When conducting a series of trials it is sometimes desirable to know the most likely
outcome. For example, what is the most likely number of acceptable components in a
sample of five tested?
Example29.19 Theprobabilityacomponentisacceptableis0.8.Fivecomponentsarepickedatrandom.
What isthe mostlikely number of acceptable components?
( 5
Solution P(no acceptable components) = (0.8)
0)
0 (0.2) 5 = 3.2 ×10 −4
( 5
P(1 acceptable component) = (0.8)
1)
1 (0.2) 4 = 6.4 ×10 −3
( 5
P(2 acceptable components) = (0.8)
2)
2 (0.2) 3 = 0.0512
( 5
P(3 acceptable components) = (0.8)
3)
3 (0.2) 2 = 0.2048
( 5
P(4 acceptable components) = (0.8)
4)
4 (0.2) 1 = 0.4096
( 5
P(5 acceptable components) = (0.8)
5)
5 (0.2) 0 = 0.3277
The mostlikely number of acceptable components isfour.
Example 29.19 illustrates an important general result. Suppose we conductnBernoulli
trialsandwishtofindthemostlikelynumberofsuccesses.Ifp =probabilityofsuccess
on a single trial,andi =mostlikely number ofsuccesses inntrials,then
p(n+1)−1<i<p(n+1)
InExample 29.19, p = 0.8,n = 5 and so
(0.8)(6) −1 <i < (0.8)(6)
3.8<i<4.8
Sinceiisan integer, theni = 4.
EXERCISES29.10
1 Theprobabilityacomponentisacceptableis0.8.Four
components are sampled. What is the probability that
(a) exactly one is acceptable
(b) exactly two are acceptable?
2 Amachine requires allseven ofitsmicro-chips to
operate correctly in order to be acceptable. The
probability amicro-chip isoperating correctly is 0.99.
(a) What isthe probability the machine is
acceptable?
(b) What isthe probability that sixofthe seven chips
are operating correctly?
(c) Themachine is redesigned sothat the original
seven chips are replaced by four new chips.The
probability a new chipoperates correctly is0.98.