Question

This problem demonstrates the concept of redundancy, the principle of improving reliability of systems by using redundant or backup components. Suppose a hydraulic system on an airplane has a 5% probability of failing.

a) What is the probability that the hydraulic system does not fail (that is, the flight is completed safely)?

b) Now, suppose that the airplane has three independent hydraulic systems that each have a 5% probability of failure. What is the probability that all three systems fail?

c) Supposing that the airplane will safely complete its flight if at least one of the three independent hydraulic systems is functioning, what is the probability that the airplane will safely complete its flight?

d) What effect does having a redundancy of three independent hydraulic systems have on the probability of a safe flight?

Answer 1

S : Event of Hydraulic system Failing

P(S) = probability of Hydraulic system Failing = 5/100 = 0.05

: Event of Hydraulic system not Failing

a)

Probability that the hydraulic system does not fail = P()

P() = 1-P(S) = 1-0.05 = 0.95

Probability that the hydraulic system does not fail = P() = 0.95

Probability that the airplane will safely complete its flight = 0.95

b)

Airplane has three independent hydraulic systems that each have a 5% probability of failure.

Let

S1: Event of Hydraulic system 1 Failing

S2: Event of Hydraulic system 2 Failing

S3 :  Event of Hydraulic system 3 Failing

P(S1)=P(S2)=P(S3) = 0.05

Probability that all three systems fail : P(F) = P(S1 and S2 and S3) = P(S1)P(S2)P(S3) = 0.05 x 0.05 x 0.05 = 0.000125

c)

Probability that the airplane will safely complete its flight

= Probability that at least at least one of the three independent hydraulic systems is functioning

= 1 - Probability that all three systems fail

= 1 - P(F) = 1- 0.000125 = 0.999875

Probability that the airplane will safely complete its flight = 0.999875

d)

From (a) With only one hydraulic system, Probability that the airplane will safely complete its flight = 0.95

From (c) , having a redundancy of three independent hydraulic systems, Probability that the airplane will safely complete its flight = 0.999875

The effect of  having a redundancy of three independent hydraulic systems is that the probability of safe flight has increased from 0.95 to 0.999875;