A mechanical system with 5 pump assemblies will operate satisfactorily if any th

Question

A mechanical system with 5 pump assemblies will operate satisfactorily if any three out of five pumps are functioning. Given that 15% of the total pump production is defective, what is the probability of system malfunction? Past data indicated that there were on an average 4 accidents on a highway per year. Number of accidents per year may be assumed to have Poisson distribution. The mean of Poisson distribution, is given by Theta. Find the probability of 1) no accidents; 2) 4 accidents, 3) at least 4 accidents per year. A light bulb (the lifetime is assumed to follow an exponential distribution) has a mean life of 400 hours. What is the probability of the bulb lasting 1) less than 300 hours; 2) more than 500 hours; 3) between 200 and 500 hours? The acceptable tolerance on a shaft product is 9 plusminus 0.005 in. From past experience, it is known that sigma = 0.003 in. What is the percentage of scrap parts? Determine the normal population % that lies within plusminus 1.0 sigma, plusminus 2.0 sigma, plusminus 3.0 sigma, plusminus 4.0 sigma.

Explanation / Answer

1) Let X be random variable which represents number of pump assemblies which are not in functioning out of five pumps & probability of individual pump will not in function is p=15/100=0.15 .Clearly , X follows Binomial distribution with parameters n=5 & p=0.15

Therefore, probability mass function of X is ,

P(X=x)=nCx px (1-p)n-x=5Cx 0.15x 0.855-x ; x=0,1,2,3,4,5

We have to find here probability of system malfunction.Note that system malfunction will occure if atleast three pump will not in function .This probability is equivalent to probability of atleast three pump will not in function.Let us find it.

P(X>=3)=P(X=3) +P(X=4) +P(X=5)

= 5C3 0.153 0.855-3 + 5C4 0.154 0.855-4 + 5C5 0.155 0.855-5

={5!/[3!*(5-3)!]}153 0.852 + {5!/[4!*(5-4)!]} 0.154 0.85 +{5!/[5!*(5-5)!]} 0.155 0.850

={5*4*3!/[3!*2*1]} 0.153 0.852 + {5*4!/[4!*1!]} 0.154 0.85 +{5!/[5!*0!]} 0.155

=10* 0.153 0.852 +5*0.154 0.85 +0.155   Since 1!=0!=1

=0.02438438 + 0.002151562+7.59375e-05

   =0.02661188

Thus, probability of system malfunction is 0.02661188 .

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