Answer to Question #163278 in Statistics and Probability for Dush

Question #163278

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.


1
Expert's answer
2021-02-24T06:53:07-0500

The probability, that the system will fail:

"= P( E_1 \\cap E_2 \\cap E_3 \\cap E_4 )"

E1 = component 1 fails

E2 = component 2, 3 fail

E3 = component 2, 4 fail

E4 = component 3, 4 fail

"P(E_1) = f_1 \\\\\n\nP(E_2) = f_2 \\times f_3 \\\\\n\nP(E_3) = f_2 \\times f_4 \\\\\n\nP(E_4) = f_3 \\times f_4"

The probability, that the system will fail:

"= P(E_1) + P(E_2) + P(E_3) + P(E_4) -P( E_2 \\cap E_3E_4 ) \\\\\n= f_1 + f_2 \\times f_3 + f_2 \\times f_4 + f_3 \\times f_4 \u2013 f_2 \\times f_3 \\times f_4"


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