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2-2 INTERPRETATIONS OF PROBABILITY 31

Axioms of
Probability Probability is a number that is assigned to each member of a collection of events
from a random experiment that satisfies the following properties:
If S is the sample space and E is any event in a random experiment,

(1) P1S2 1
(2) 0 P1E2 1
(3) For two events E and E with E ¨ E
1
2
1
2
´ E 2 P1E 2 P1E 2
P1E 1 2 1 2

The property that 0 P1E2 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 rel-
ative frequency of S is 1. Property 3 implies that if the events E and E have no outcomes in
2
1
common, the relative frequency of outcomes in E ´ E 2 is the sum of the relative frequencies
1
of the outcomes in E and E .
2
1
These axioms imply the following results. The derivations are left as exercises at the end
of this section. Now,
P1 2 0

and for any event E,

P1E¿2 1 P1E2

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 is contained
1
in the event E ,
2
P1E 2 P1E 2
1
2
EXERCISES FOR SECTION 2-2
2-34. Each of the possible five outcomes of a random ex- 2-36. A part selected for testing is equally likely to have
periment is equally likely. The sample space is {a, b, c, d, e}. been produced on any one of six cutting tools.
Let A denote the event {a, b}, and let B denote the event (a) What is the sample space?
{c, d, e}. Determine the following: (b) What is the probability that the part is from tool 1?
(a) P1A2 (b) P1B2 (c) What is the probability that the part is from tool 3 or
(c) P1A¿2 (d) P1A ´ B2 tool 5?
(e) P1A ¨ B2 (d) What is the probability that the part is not from tool 4?
2-35. The sample space of a random experiment is {a, b, c, 2-37. An injection-molded part is equally likely to be ob-
d, e} with probabilities 0.1, 0.1, 0.2, 0.4, and 0.2, respectively. tained from any one of the eight cavities on a mold.
Let A denote the event {a, b, c}, and let B denote the event (a) What is the sample space?
{c, d, e}. Determine the following: (b) What is the probability a part is from cavity 1 or 2?
(a) P1A2 (b) P1B2 (c) What is the probability that a part is neither from cavity 3
(c) P1A¿2 (d) P1A ´ B2 nor 4?
(e) P1A ¨ B2