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A machine operates only when all of its three components function. The probabilities of the failures of the first, second and third components are 0.14, 0.10 and 0.05, respectively. What is the probability that the machine will fail?

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Given: let A ,B and C be the three components of a machine which works only if all its three compenents function.

the probabilities of the failures of A,B and C respectively is given i.e, 

P(A) = 0.14 ,P(B) = 0.10 and P(C) = 0.05

\(\Rightarrow\) P(\(\overline A\)) = = 0.86 and P(\(\overline B\)) = 0.90 and P(\(\overline C\)) = 0.95

To Find: The probability that the machine will fail.

Here, P(the machine will fail) = 1 – P(the machine will function)

= 1 – P(all three components are working)

Therefore, The probability that the machine will fail is 0.2647.

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