Three components, one of them essential

Topic: Reliability · more questions on this topic
Structure of a k-out-of-n system ›

Components 1, 2 and 3 fail independently with probability 0.1 each. The system works when component 1 works and at least one of components 2 and 3 works. Find the probability that the system works.

Let A_i be the event that component i works, so \mathbb{P}(A_i)=0.9. The system works on A_1\cap(A_2\cup A_3). Independence gives

\mathbb{P}(A_2\cup A_3)=0.9+0.9-0.9^2=0.99,

and the system probability is 0.9\times 0.99=0.891.