30 2. FUNDAMENTAL RELIABILITY MATHEMATICS
e probability of the safe machine unit will be:
P
.
B
/
D
25;192 1053
25;192
D 0:958 D 95:8%:
2.4.2 AXIOMATIC DEFINITION
In the axiomatic approach to defining probability, it is assumed that the occurrence of every
sample point in an experiment will have the same likelihood. For an example of tossing a coin, the
occurrence of the likelihood of “head” or “tail” is the same. For another example, the occurrence
of showing number 1 in rolling a dice will have the same likelihood as the occurrence of showing
number 6.
According to the axiomatic approach, the probability of occurrence of an event E, denoted
as P .E/, is defined as a numerical value of P
.
E
/
such that P.E/ obeys the following three
axioms or postulates.
1. e probability of an event E cannot be a negative value because it has not any physical
meaning. e event E can happen with a positive probability or cannot happen with a
zero probability, that is,
P
.
E
/
0: (2.11)
2. If E is a universal set , the occurrence of a universal set is a certain event, that is
P
.
/
D 1: (2.12)
3. If events E
1
and E
2
are mutually exclusive, the probability of the union of two events is
equal to the sum of the probability of each mutually exclusive event, that is
P
.
E
1
[ E
2
/
D P
.
E
1
/
C P
.
E
2
/
: (2.13)
Example 2.7
e experiment of rolling a dice will have six possible sample point or outcome, that is, number
1, 2, 3, 4, 5, or 6. (1) Calculate the probability of showing number 5. (2) If the set A D
f
3; 5
g
,
calculate the probability of the set A.
Solution:
1. Calculate the probability of showing number 5.
2.4. DEFINITION OF PROBABILITY 31
e universal set of rolling dice is D
f
1; 2; 3; 4; 5; 6
g
. ere are six sample points in the
universal set. Since all of the sample points are a mutually exclusive event, we have:
P
.
/
D 1 D P .
f
1; 2; 3; 4; 5; 6
g
/
D P
.
f
1
g
/
C P
.
f
2
g
/
C P
.
f
3
g
/
C P
.
f
4
g
/
C P
.
f
5
g
/
C P
.
f
6
g
/
:
Since each sample point will have the same occurrence likelihood, we have
P
.
f
1
g
/
D P
.
f
2
g
/
D P
.
f
3
g
/
D P
.
f
4
g
/
D P
.
f
5
g
/
D P
.
f
6
g
/
:
erefore, P
.
f
1
g
/
D P
.
f
2
g
/
D P
.
f
3
g
/
D P
.
f
4
g
/
D P
.
f
5
g
/
D P
.
f
6
g
/
D
1
6
.
2. A D
f
3; 5
g
, calculate P .A/.
P
.
A
/
D P
.
f
3
g
/
C P
.
f
5
g
/
D
1
6
C
1
6
D
1
3
:
Example 2.8
ere are three white balls and four red balls in a box. e experiment of randomly picking
one ball from the box is conducted for a total of 500 times. Among these 500 experiments, we
get 203 white balls and 397 red balls. (1) Use the relative frequency definition to calculate the
probability of picking a white ball. (2) Use the axiomatic definition to calculate the probability
of picking a white ball.
Solution:
• P
.
whiteball
/
by the relative frequency definition.
According to the relative frequency definition, we have
P
.
whiteball
/
D
203
500
D 0:406 D 40:6%:
• P
.
whiteball
/
by the axiomatic definition.
According to the axiomatic definition, we can have
P
.
whiteball
/
D
3
7
D 0:429 D 42:9%:
ere is some difference between the two results obtained above. When we calculate the
probability of an event by the axiomatic definition, we do not need to do any actual exper-
iment. We need to calculate a number of all possible sample points (outcomes) and run
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