Probability of system failure

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 $f_i$ for $i = 1,\dots, 4$. 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 sample space is a discrete set of $2^4$ possibilities. To get the probability of failure it is simplest to just list the disjoint combinations that lead to failure. These are 1 fails and the rest don't matter and 1 succeeds and any of the following occur: 2,3 fail and 4 succeeds; 2,4 fail and 3 succeeds; 3,4 fail and 2 succeeds; and 2,3,4 fail. The probability of failure is then ${f_1+(1-f_1)(f_zf_3(1-f_4)+f_2f_4(1-f_3)+f_3f_4(1-f_2)+f_2f_3f_4)}$

${=f_1+(1-f_1)(f_2f_3+f_2f_4+f_3f_4-2f_2f_3f_4).}$

Then main point is insuring that there is no overlap when forming the sum of probabilities. The second form for the answer is to simplify the arithmetic.