Question

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 1

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

$P(E_1) = f_1 \\ P(E_2) = f_2 \times f_3 \\ P(E_3) = f_2 \times f_4 \\ P(E_4) = f_3 \times f_4$

The probability, that the system will fail:

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