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If pip_ipi​​ denote the reliability of the ithi^{th}ith component; i=1,2,…,n i=1,2,\ldots,ni=1,2,…,n​ then the reliability of parallel
Question

If pip_i​ denote the reliability of the ithi^{th} component; i=1,2,…,n i=1,2,\ldots,n​ then the reliability of parallel system is given by

A.

​∏i=1npi \prod_{i=1}^{n}p_i​​

B.

​1−∏i=1n(1−pi)1-\prod_{i=1}^{n}(1-p_i)​​

C.

​∏i=1n(1−pi)\prod_{i=1}^{n}(1-p_i)​​

D.

​1−∏i=1npi1-\prod_{i=1}^{n}p_i​​

Correct option is B

Given:​
Reliability of the ithi^{th}​​ component= pi.p_i.​​
Formula Used:
Reliability of a parallel system
=1−∏i=1n(1−pi).=1-\prod_{i=1}^{n}(1-p_i).​​​
Solution:
A parallel system fails only when all its components fail.
Probability that the ithi^{th}​ component fails= 1−pi.1-p_i.​​
Hence,
​P(all components fail)=∏i=1n(1−pi).P(\text{all components fail})=\prod_{i=1}^{n}(1-p_i).​​
Therefore,
Reliability of the parallel system
​=1−∏i=1n(1−pi).=1-\prod_{i=1}^{n}(1-p_i).​​
The correct answer is (b) 1−∏i=1n(1−pi).1-\prod_{i=1}^{n}(1-p_i).​​

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