Fundamentals of Probability and Statistics for Engineers

2.10 For the two components described in Problem 2.9, tests have produced the follow-
ing result:

Determine the probability that:

(a) The second component is good.

(b) At least one of the components is good.

(c) The first component is good given that the second is good.

(d) The first component is good given that at most one component is good.

ForthetwoeventsAandB:

(e) Are they independent? Verify your answer.

(f) Are they mutually exclusive? Verify your answer.

2.11 A satellite can fail for many possible reasons, two of which are computer failure
and engine failure. For a given mission, it is known that:

The probability of engine failure is 0.008.

The probability of computer failure is 0.001.

Given engine failure, the probability of satellite failure is 0.98.

Given computer failure, the probability of satellite failure is 0.45.

Given any other component failure, the probability of satellite failure is zero.

(a) Determine the probability that a satellite fails.

(b) Determine the probability that a satellite fails and is due to engine

failure.

(c) Assume that engines in different satellites perform independently. G iven a

satellite has failed as a result of engine failure, what is theprobability that

the same will happen to another satellite?

2.12 Verify Equation (2.14).

2.13 Show that, for arbitrary events

This is known as Boole’s inequality.

2.14 A box contains 20 parts, of which 5 are defective. Two parts are drawn at random
from the box. What is the probability that:
(a) Both are good?
(b) Both are defective?
(c) One is good and one is defective?

2.15 An automobile braking device consists of three subsystems, all of which must work
for the device to work. These systems are an electronic system, a hydraulic system,
and a mechanical activator. In braking, the reliabilities (probabilities of success) of
these units are 0.96, 0.95, and 0.95, respectively. Estimate the system reliability
assuming that these subsystems function independently.
Comment: systems of this type can be graphically represented as shown in
Figure 2.10, in which subsystems A (electronic system), B (h ydraulic system), and

30 Fundamentals of Probability and Statistics for Engineers

P…A†ˆ 0 : 8 ; P…BjA†ˆ 0 : 85 ; P…BjA†ˆ 0 : 75 :

A 1 ,A 2 ,...,An,

P…A 1 [A 2 [...[An†P…A 1 †‡P…A 2 †‡‡P…An†