SOLUTION: An electronic system has four components labeled as 1, 2, 3, and 4. The system has to be used during a given time period. The probability that component 𝑖 will fail during t

Question 1197515: An electronic system has four components labeled as 1, 2, 3, and 4. The system has to
be used during a given time period. The probability that component 𝑖 will fail during
that time period is Fi
for 𝑖 = 1, …, 4. Failures of the components are physically
independent of each other. A system failure occurs if component 1 fails or if at least
two of the other components fail. Specify an appropriate sample space and determine
the probability of a system failure.
Answer by ikleyn(52776)   (Show Source): You can put this solution on YOUR website!
.
An electronic system has four components labeled as 1, 2, 3, and 4. The system has to
be used during a given time period. The probability that component 𝑖 will fail during
that time period is Fi
for 𝑖 = 1, …, 4. Failures of the components are physically
independent of each other. A system failure occurs if component 1 fails or if at least
two of the other components fail. Specify an appropriate sample space and determine
the probability of a system failure.
~~~~~~~~~~~~~~~~~

        If the final goal is to determine the probability for the whole system to fail, then
        it is much simpler to solve the problem without considering the appropriate sample space.

        Formula (1) below replaces considering sample space.

If    is the probability to fail for i-th component,
then the probability not to fail for this component is  = 1 - .


Now the probability for the whole system to fail is

    P = P(1st component failed) + P(1st is good, 2nd failed,  3rd failed,  4th is good)   + 
                                + P(1st is good, 2nd failed,  3rd is good, 4th failed)    + 
                                + P(1st is good, 2nd is good, 3rd failed,  4th failed)    + 
                                + P(1st is good, 2nd failed,  3rd failed,  4th failed).      (1)


It is the same as


    P =  +  +  = 


      =  +  + .    ANSWER

Solved.

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