Page 55 - Applied statistics and probability for engineers

Section 2-2/Interpretations and Axioms of Probability 33

Axioms of
Probability Probability is a number that is assigned to each member of a collection of events
from a random experiment that satisies the following properties:
If S is the sample space and E is any event in a random experiment,
(1) P S ( ) = 1
(2) 0 ≤ ( ) ≤P E 1
with E 1 ∩ E 2 = ∅
(3) For two events E 1 and E 2
P E 1) + ( )
P E 1 ∪ ) = ( P E 2
(
E 2
The property that 0 ≤ P E( ) ≤ 1 is equivalent to the requirement that a relative frequency must be
between 0 and 1. The property that P S( ) = 1 is a consequence of the fact that an outcome from the
sample space occurs on every trial of an experiment. Consequently, the relative frequency of S is 1.
Property 3 implies that if the events E 1 and E 2 have no outcomes in common, the relative frequency
of outcomes in E 1 ∪ E 2 is the sum of the relative frequencies of the outcomes in E 1 and E 2 .
These axioms imply the following results. The derivations are left as exercises at the end
of this section. Now,
P ∅ ( ) = 0
and for any event E,
P E)
P E′ ( ) = − (
1
For example, if the probability of the event E is 0.4, our interpretation of relative frequency implies
that the probability of E′ is 0.6. Furthermore, if the event E 1 is contained in the event E 2 ,
P E 1 ( ) ≤ ( )
P E 2

Exercises FOR SECTION 2-2

Problem available in WileyPLUS at instructor’s discretion.
Tutoring problem available in WileyPLUS at instructor’s discretion

2-58. Each of the possible ive outcomes of a random experi- (b) What is the probability that an order does not request more
ment is equally likely. The sample space is { , , , , }a b c d e . Let A than one optional feature?
denote the event { , }a b , and let B denote the event { , , }c d e . Deter- 2-61. If the last digit of a weight measurement is equally
mine the following: likely to be any of the digits 0 through 9,
(a) P A( ) (b) P B( ) (c) P A( )′ (a) What is the probability that the last digit is 0?
(d) P A( ∪ B) (e) P A( ∩ B) (b) What is the probability that the last digit is greater than or
equal to 5?
2-59. The sample space of a random experiment is { , ,a b
,
,
c d e} with probabilities 0.1, 0.1, 0.2, 0.4, and 0.2, respec- 2-62. A part selected for testing is equally likely to have been
tively. Let A denote the event { , , }a b c , and let B denote the produced on any one of six cutting tools.
(a) What is the sample space?
event { , , }c d e . Determine the following:
(a) P A( ) (b) P B( ) (c) P A( ′) (b) What is the probability that the part is from tool 1?
(c) What is the probability that the part is from tool 3 or tool 5?
(d) P A( ∪ B) (e) P A( ∩ B) (d) What is the probability that the part is not from tool 4?
2-60. Orders for a computer are summarized by the 2-63. An injection-molded part is equally likely to be obtained
optional features that are requested as follows: from any one of the eight cavities on a mold.
(a) What is the sample space?
Proportion of Orders
(b) What is the probability that a part is from cavity 1 or 2?
No optional features 0.3
(c) What is the probability that a part is from neither cavity 3 nor 4?
One optional feature 0.5
2-64. In an acid-base titration, a base or acid is gradually
More than one optional feature 0.2 added to the other until they have completely neutralized each
(a) What is the probability that an order requests at least one other. Because acids and bases are usually colorless (as are the
optional feature? water and salt produced in the neutralization reaction), pH is