Problem 157E

The following circuit operates if and only if there is a path of functional devices from left to right. The probability that each device functions is as shown. Assume that the probability that a device is functional does not depend on whether or not other devices are functional. What is the probability that the circuit operates?

Solution :

Step 1 of 1:

Given if and only if circuit a path of functional device from left to right.

Then the diagram is given below.

Our goal is:

We need to find the probability that the circuit operates.

Now we have to find the probability that the circuit operates.

From the above circuit we have to calculate P(), P(), P(), P() , P(), and P().

P(A) = P()+P()-P ()

P(A) = P()+P()-P()P()

P(A) = 0.9+0.95-(0.90.95)

P(A) = 1.85-0.855

P(A) = 0.995

Therefore, P(A)=0.995.

P(B) = P()+P()-P ()

P(B) = P()+P()-P()P()

P(B) = 0.9+0.95-(0.90.95)

P(B) = 1.85-0.855

P(B) = 0.995

Therefore, P(B)=0.995.

P(C) = P()+P()-P ()

P(C) = P()+P()-P()P()

P(C) = 0.8+0.9-(0.80.9)

P(C) = 0.98

Therefore, P(C)=0.98.

The probability that the circuit operates is

P(A)P(B)P(C) = 0.9950.9950.98

P(A)P(B)P(C) = 0.9702

Therefore, the probability that the circuit operates is 0.9702.