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ERBoard Learn Dhttps://webcourses.niu.edu x x cswebdavl pid-3831611-dt-content-rid-23001326 2/courses/20152-ISYE-335 ----11Hw9%234.pdf 6. The manufacturing of semiconductor chips produces 2% defective chips. Assume the chips are independent and that a lot contains 1000 chips. (a) Approximate the probability that more than 25 chips are defective. (b) Approximate the probability that between 20 and 30 chips are defective. 7. The life of a semiconductor laser at a constant power is normally distributed with a mean of 7000 hours and a sta dard deviation of 600 hours. (a) What is probability that a laser fails before 5000 hours? (b) What is the life in hours that 95% of the lasers exceed? (c) If three lasers are used in a product and they are assumed to fail independently, what is the probability that all three are still operating after 7000 ours?

Solution

(6)(a) Given X follows Binomial distribution with n=1000 and p=0.02

P(X=x)=1000Cx*(0.02^x)*(0.98^(1000-x)) for x=0,1,2,...,1000

So the probability is

P(X>25) = P(X=26)+P(X=27)+...+P(X=1000)

=1000C26*(0.02^26)*(0.98^(1000-26))+...+1000C1000*(0.02^1000)*(0.98^(1000-1000))

=0.1099331

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(b) P(20<=X<=30) = P(X=20)+P(X=21)+...+P(X=30)=0.5179937

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(7)(a)P(X<5000) = P((X-mean)/s <(5000-7000)/600)

=P(Z<-3.33) =0.0004 (from standard normal table)

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(b) P(X<c)=0.05

--> P(Z<(c-7000)/600) =0.05

--> (c-7000)/600 =-1.64 (from standard normal table)

So c= 7000-1.64*600=6016

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(c) The probability that are operating after 7000 hours is

P(X>7000) = P(Z>(7000-7000)/600) = P(Z>0)=0.5 (from standard normal table)

So all three is

0.5^3=0.125