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

Denote by A_i the event that component i will fail during that time period.

Then the event that a system failure occurs can be expressed by the formula:

$A_1\vee\bar A_1(A_2A_3\bar A_4\vee\bar A_2 A_3A_4\vee A_2\bar A_3A_4\vee A_2 A_3 A_4)$

All the terms are disjoint, therefore the probability of this event is a sum of the probabilities of each component. Sinse the events $A_1, A_2, A_3, A_4$ are independent, the probability of the events which are intersections of them (or of their negatives) is a product of the individual probabilities. Finally,

$P(A_1\vee\bar A_1(A_2A_3\bar A_4\vee\bar A_2 A_3A_4\vee A_2\bar A_3A_4\vee A_2 A_3 A_4))=f_1+(1-f_1)(f_2f_3(1-f_4)+(1-f_2)f_3f_4+f_2f_4(1-f_3)+f_2f_3f_4)=f_1+(1-f_1)(f_2f_3+f_3f_4+f_2f_4+2f_2f_3f_4)$

if all the values f₂,f₃ and f₄ are non-zero, this probability is equal to

$f_1+f_2f_3f_4(1-f_1)(2+f_2^{-1}+f_3^{-1}+f_4^{-1})$