An engineering system consisting of n components

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SUMMARY

The discussion focuses on calculating the conditional probability of component 1 functioning in a k-out-of-n system, specifically when k=2 and n=3, with each component having a probability of 1/2 of functioning independently. The solution utilizes binomial probability to determine P(B), the probability that at least 2 out of 3 components function, resulting in P(B) = 3/8. The conditional probability P(A|B) is calculated as 1/2, confirming the correctness of the solution provided, which does not require considering the scenario where all components function since it does not affect the outcome for k=2.

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TomJerry
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Problem:
An engineering system consisting of n components is said to be a k-out-of-n system (k<= n)if and only if at least k of the n components function.Suppose all the components function independently of each other with a probability 1/2. Find the conditional probability that component 1 is working given that the system functions, when k=2 and n=3.

Solution:
Let A be component 1 functions i.e P(A) = 1/2
Let B be that 2 of 3 components function i.e P(B) = P(x=2;n=3;p=1/2) = 3*1/4*1/2 = 3/8 [using binomial probability]
Therefore
P(A/B) = P(A intersection B) / P(B) = P(A) * P(B) / P(B) = (1/2 * 3/8) / (3/8) = 1/2 [IS THIS SOLUTIONS CORRECT]
 
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I saw this on another forum. You need to include possibility that all 3 components function.
 

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