0
$\begingroup$

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.

$\endgroup$
0
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.