Question

A k out of n system is one in which there is a group of n components, and the system will function if at least k of the components function. Assume the components function independently of one another.

a) In a 3 out of 5 system, each component has probability 0.6 of functioning. What is the probability that the system will function? Round the answer to six decimal places.

b) In a 3 out of n system, in which each component has probability 0.6 of functioning, what is the smallest value of n needed so that the probability that the system functions is at least 0.90?

Answer 1

a)

here this is binomial with parameter n=5 and p=0.6

e probability that the system will function =P(3 or more will function)

P(X>=3)=1-P(X<=2)=

1-∑x=02 (nCx)px(q)(n-x) =

0.682560

b)

here P(system functions) 1-∑_x=0² (_nC_x)p^x(q)^(n-x) >=0.90

by hit and trail method:

for n=6 ; p=0.820800

for n=7 ; p=0.903744

since condition is fulfilled for n=7, smallest value of n=7