Assume C and D function independently. For the system to function, either C or D must function.
A system contains two components, C and D, connected in parallel as shown in the diagram.
a. If the probability that C fails is 0.08 and the probability that D fails is 0.12, find the probability that the system functions.
b. If both C and D have probability p of failing, what must the value of p be so that the probability that the system functions is 0.99?
c. If three components are connected in parallel, function independently, and each has probability p of failing, what must the value of p be so that the probability that the system functions is 0.99?
d. If components function independently, and each component has probability 0.5 of failing, what is the minimum number of components that must be connected in parallel so that the probability that the system functions is at least 0.99?
Let C denote the event that component C functions, and let D denote the event that component D functions.
(a)
Alternatively,
(b) P(system functions) =
(c) P(system functions) = 
= 0.99. Therefore 
= 0.2154
(d) Let n be the required number of components. Then n is the smallest integer such that 
? 0.99. It follows that n ln(0.5) ? ln 0.01, so n ? (ln 0.01)(ln 0.5) = 6.64. Since n must be an integer, n = 7.
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