Question

Suppose that magnetic tape made by a certain factory has an average of 1 flaw per 60 meters of tape. If the distribution of these flaws is modeled by a Poisson process, find (a) the probability that a 100m section of tape will have at most 1 flaw. (b) the probability that a 75 meter section will have 3 or more flaws. (c) the maximum length of a section for which the probability that it will contain no flaws is at least 90%.

Answer 1

a)expected number of flaw in 100m section =t =(1/60)*100 =5/3

probability that a 100m section of tape will have at most 1 flaw :

P(X<=1) =P(X=0)+P(X=1) =e^-5/3(5/3)⁰/0! +e^-5/3(5/3)¹/1! =0.5037

b)

expected number of flaw in 75m section =t =(1/60)*75 =1.25

probability =

P(X>=3)=

1-P(X<=2)=

1-∑x=02{e-λ*λx/x!}=

0.1315

c)

let length is t

therefore P( 0 flaws) e^-t >0.90

t =-ln(0.90)*60 =6.3216 m