Assume A, B, C, and D function independently. If the probabilities that A, B, C, and D fail are 0.1, 0.2, 0.05, and 0.3, respectively, what is the probability that the system functions?
A system consists of four components connected as shown.
The system functions if at least one of the four components A, B, C, D, function. Therefore P(system functions) = 1 – P(all components fail). Let A be the event that component A functions, let ? be the event that component ? functions, let C be the event that component C functions, and let D be the event that component D functions. Then P(system functions) = 
= 1 – (0.1)(0.2)(0.05)(0.3) = 0.9997
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