Q.3.53 - A First Course in Probability

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 $1 2$ . Find the conditional probability that component 1 works given that the system is functioning. 

Step-by-Step Solution

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 1: 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 

Step 2: 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.

Step 3: 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,

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

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

Step 4: Final Answer

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