A system composed of n separate components is said to be parallel system if it functions when at least one of the components functions. For such a system, if a component i functions with probability pi, independent of other components, i = 1, 2,....n, what is the probability that the system functions?a)p1p2...pnb)p1 + p2 + ... pnc)1 - (1-p1)(1 - p2) ... (1-pn)d)(1-p1)(1 - p2) ... (1-pn)Correct answer is option 'C'. Can you explain this answer?

Question

Answer

Understanding the Problem:
We are given a system composed of n separate components. The system is said to be a parallel system if it functions when at least one of the components functions. Each component functions independently with a certain probability pi, where i = 1, 2, ..., n. We need to find the probability that the system functions.

Solution:

To find the probability that the system functions, we need to consider the cases where at least one component functions.

Case 1: Only one component functions.
In this case, we can choose any one of the n components to function, and the remaining n-1 components should not function. The probability of this case is given by:
P(Only one component functions) = n * p1 * (1-p2) * (1-p3) * ... * (1-pn)

Case 2: Two components function.
In this case, we can choose any two components to function, and the remaining n-2 components should not function. The probability of this case is given by:
P(Two components function) = (nC2) * p1 * p2 * (1-p3) * ... * (1-pn)

Case 3: Three components function.
In this case, we can choose any three components to function, and the remaining n-3 components should not function. The probability of this case is given by:
P(Three components function) = (nC3) * p1 * p2 * p3 * (1-p4) * ... * (1-pn)

Case n: All n components function.
In this case, all the n components should function. The probability of this case is given by:
P(All n components function) = p1 * p2 * p3 * ... * pn

The total probability that the system functions is the sum of probabilities of all these cases. Therefore, the final probability is given by:
P(System functions) = P(Only one component functions) + P(Two components function) + ... + P(All n components function)

Simplifying this expression, we get:
P(System functions) = n * p1 * (1-p2) * (1-p3) * ... * (1-pn) + (nC2) * p1 * p2 * (1-p3) * ... * (1-pn) + (nC3) * p1 * p2 * p3 * (1-p4) * ... * (1-pn) + ... + p1 * p2 * p3 * ... * pn

This can be further simplified as:
P(System functions) = 1 - (1-p1)(1-p2)(1-p3)...(1-pn)

Therefore, option C, (1-p1)(1-p2)...(1-pn), is the correct answer.

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