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 .2 of failing in less than 1000 hours. The subsystem will operate if any two of the four components are operating. Assume that the components operate independently. Find the probability that

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

b the subsystem operates longer than 1000 hours.

Step 1 of 2:

Given 4 identical components are there in one subsystem.

Then the probability of 0.2 of failing in less than 1000 hours.

We assume that the components operate are independent.

Our goal is:

a). We need to find the probability that exactly two of the four components last longer than

1000 hours.

b). We need to find the probability that the subsystem operates longer than 1000 hours.

a).

Now we have to find the probability that exactly two of the four components last longer than 1000 hours.

Let Y is the number of components failing in less than 1000 hours.

Here Y is binomial with parameter n and p.

So n=4 and p=0.8 and

q=1-p

q=1-0.20

q=0.80

Therefore q=0.80.

Then the probability that exactly two of the four components last longer than 1000 hours is

The formula of the binomial is

P(Y) =

Then, P(Y=2) is

P(Y=2)=

P(Y=2)=

P(Y=2)=

P(Y=2)= 0.1536

Therefore, the probability that exactly two of the four components last longer than 1000 hours is 0.1536.