A complex electronic system is built with a certain number o

A complex electronic system is built with a certain number of backup components in its subsystems. One subsystem has four identical components, each with a probability of 0.15 of failing in less than 1000 hours. The subsystem will operate if any two or more of the four components are operating. Assuming that the components operate independently, find the probabilities of the following events.

a Exactly two of the four components last longer than 1000 hours.

b The subsystem operates for longer than 1000 hours.

Solution

There are 4 identical components

prob of failing in less than 1000 hours =0.15

All components work independently

Hence X no of components failing in less than 1000 hours is binomial with p =0.15 and n = 4

a) Prob Exactly two of the four components last longer than 1000 hours.

= P(two other components fail)

= P(X=2)

= 0.0975

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B) The subsystem operates for longer than 1000 hours

= P(2 or more working)

= P(1 or less failing)

= P(X<=1)

= 0.8905