Question

A k out of n system is one in which there is a group of ncomponents, and the system will function if at least kof the components function. Assume the components function independently of one another. In a 3 out of 5 system, each component has a probability of 0.9 of functioning. What is the probability that system will function?

Answer 1

Answer:

0.99145 is the probability that the system will work properly.

Step-by-step explanation:

We are given the following information:

The components function independently of one another.

We treat component working properly as a success.

P(component working properly) = 0.9

Then the number of components follows a binomial distribution, where

$P(X=x) = \binom{n}{x}.p^x.(1-p)^{n-x}$

where n is the total number of observations, x is the number of success, p is the probability of success.

Now, we are given n = 5 and x = 3

For the system to function if atleast 3 out of 5 will work properly.

We have to evaluate:

$P(x \geq 3) = 1 - P(x = 1) - P(x = 2) \\=1- \binom{5}{1}(0.9)^1(1-0.9)^4 - \binom{5}{2}(0.9)^2(1-0.9)^3\\=1-0.00045-0.0081\\= 0.99145$

0.99145 is the probability that the system will work properly.