Applied Statistics and Probability for Engineers

2-2 INTERPRETATIONS OF PROBABILITY 31

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 Sis the sample space and Eis any event in a random experiment,

(1)

(2)

(3) For two events E 1 and E 2 with

P 1 E 1 ́E 22 P 1 E 12 P 1 E 22

E 1 ̈E 2   

0 P 1 E 2  1

P 1 S 2  1

Axioms of

Probability

The property that 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 Sis 1. Property 3 implies that if the events E 1 and E 2 have no outcomes in

common, the relative frequency of outcomes in 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,

and for any event E,

For example, if the probability of the event Eis 0.4, our interpretation of relative

frequency implies that the probability of is 0.6. Furthermore, if the event E 1 is contained

in the eventE 2 ,

EXERCISES FOR SECTION 2-2

P 1 E 12 P 1 E 22

E¿

P 1 E¿ 2  1 P 1 E 2

P 1 2  0

E 1 ́E 2

0 P 1 E 2  1

2-34. Each of the possible five outcomes of a random ex-

periment is equally likely. The sample space is {a,b,c,d,e}.

Let Adenote the event {a,b}, and let Bdenote the event

{c,d,e}. Determine the following:

(a) (b)

(c) (d)

(e)

2-35. 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, respectively.

Let Adenote the event {a,b,c}, and let Bdenote the event

{c,d,e}. Determine the following:

(a) (b)

(c) (d)

(e)

2-36. A part selected for testing is equally likely to have

been produced on any one of six cutting tools.

(a) What is the sample space?

(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) What is the probability that the part is not from tool 4?

2-37. An injection-molded part is equally likely to be ob-

tained from any one of the eight cavities on a mold.

(a) What is the sample space?

(b) What is the probability a part is from cavity 1 or 2?

(c) What is the probability that a part is neither from cavity 3

nor 4?

P 1 A ̈B 2

P 1 A¿ 2 P 1 A ́B 2

P 1 A 2 P 1 B 2

P 1 A ̈B 2

P 1 A¿ 2 P 1 A ́B 2

P 1 A 2 P 1 B 2

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