Question

4. A parallel system functions whenever at least one of its components works. Suppose you have two separate parallel systems, A and B, each consisting of n identical components that work independently with probability p. a) Consider parallel system A. Given that the system is functioning, what is conditional probability that component 1 works? (This has nothing to do with B yet.) Suppose system B breaks down (all n of its components fail), but system A remains functional. To get system B running, you pick a random component from system A and swap it with a component from system B. b) What is the probability that this fixes system B? c) What is the probability that system A continues to work?

Answer 1

Answer: a) Consider paralel System A. probability =P probability that component I works P[component work]- P- P[N] Assume that: M: event thał System A works. N: event that component I WORKS. * P[MON]: 1 component i works System A must work? * P[M] - 1-P[MC] -EPI None of the component of System A is working ] 1-C1-pin as component toork independentlyy * PENIM] : P[NOM] IPCM] Telpon by what is the probability that this fixes System B? probability that this fizes System B if the Sandomly choosen Component from system of is working P[components o posting It is working] - [ - [1-p]]

P{This fines System Bla is working ] 2p [chosen Component is working 14 is working. >> Ëp [choosen Component is working lith Component is chosen A is working] =Plith Component is chosen / 4 is coorking] → Sci-[l-p]n] 'p[ith Component is chosen] {i[l-[l-p71] [As chosing ith component is independent of the fact that a is working] © What is the probability that system A continus to work } probability that System A continues to work P[System Continous to work = 1-P [system A breaks down ?imp [only the choasen Componont was working and others were not working 212 plith Component was only working and other is not working Plith Component was chosen] 1th PCI-p)n-1 // 1-PC-P31 0-Xo