A parallel system functions whenever at least one of its components works.

Question

A parallel system functions whenever at least one of its components works. Consider a system of 3 components and suppose that each component independently works with probability 0.40. Find the probability that the component 1 works given that the system is functioning.

Answer 1

Hints: You can apply the Bayes theorem. The relevant events are:

$F: \texttt{"Component 1 works"}$ and $S: \texttt{"System is functioning"}$

$$P(\texttt{F|S})=\frac{P(F\cap S )}{P(S)}$$

$F\cap S $ is the event when the system is functioning and component 1 works. Thus you need all combinations of components where at least component 1 works. These combinations are $(1,2,3);(1,2,\overline 3), (1, \overline 2, 3); (1, \overline 2, \overline 3)$

The line over the letters/numbers indicates that the corresponding component/system does not work.

And for $P(S)$ you can use the converse probability. It makes the calculation much simpler. $P(S)=1-P(\overline S)$