A system contains three components A B and C and works if an

A system contains three components, A, B, and C, and works if and only if at least two of these components work. Assume the components work independently of one another, and that the probability that a component works is P(A) = P(B) = P(C) = .7. (a) What is the probability that the system works? (b) Given that the system has failed, what is the probability that component A has failed? (Hint: Draw a Venn diagram.)

Solution

The system works if atleast two components work

Each component is independent of other

(a) Prob that the system works

= P(atleast two works)

=1-only one works

= 1-3(0.7)

= 0.79

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b) Given that system failed

P(system failed) = 1-0.79 = 0.21

Prob (A failed/system failed)

= 0.3/0.21

= 1/7