Question

A parallel system functions whenever at least one of its components works. Consider a parallel system of$n$components, and suppose that each component works independently with probability $12$. Find the conditional probability that component 1 works given that the system is functioning.

Short answer

The conditional probability that component 1 results shown that the system is functioning is$\frac{\frac{1}{2}}{1-{\left(\frac{1}{2}\right)}^n}$

Step-by-step solution

Given information 

Given in the question that, a parallel system functions whenever at least one of its components works.

We need to find the conditional probability that component 1 works given that the system is functioning

Parallel system functions 

Assume a parallel system of n components. The probability for each component to result is $p=\frac{1}{2}$

The system will operate whenever at least one component works.

The system will not function only when all the components are fell to work. For a description of probability,

$p+q=1$

$\frac{1}{2}+q=1$

$q=\frac{1}{2}$

Hence, the probability of each component that falls to perform is $q=\frac{1}{2}$

There are $n$components in the system.

Applying conditional probability

The probability that none of the functions of the components is,

$P\left(\text{none}\right)=q^n$

$={\left(\frac{1}{2}\right)}^n$

The probability that at least one of the components results is,

System function $=1-P\text{(none)}$

$=1-{\left(\frac{1}{2}\right)}^n$

The conditional probability that component 1 results shown that the system is functioning is,

$\frac{P\text{(component}1\text{works})}{P\left(\text{system functioning}\right)}$

$=\frac{\frac{1}{2}}{1-{\left(\frac{1}{2}\right)}^n}$

Final Answer 

The conditional probability that component 1 results shown that the system functioning is$\frac{\frac{1}{2}}{1-{\left(\frac{1}{2}\right)}^n}$