28 2. FUNDAMENTAL RELIABILITY MATHEMATICS
de Morgan’s rule: is rule states that the complement of unions (intersections) is equal to
the intersections (unions) of the respective complements, that is,
A [B D A B (2.7)
A B D A [ B: (2.8)
Example 2.4
In rolling a dice experiment, sets A and B are A D
f
1; 4; 6
g
and B D
f
2; 4; 5
g
. Use these two
sets to demonstrate Equations (2.7) and (2.8).
Solution:
For rolling a dice experiment, the universal set is D
f
1; 2; 3; 4; 5; 6
g
.
According to the definition of the complement of a set, we have:
A D
f
1; 4; 6
g
; A D
f
2; 3; 5
g
B D
f
2; 4; 5
g
; B D
f
1; 3; 6
g
:
According to the definition of the union of two sets, the complement of a set and intersection
of two sets, we have:
A [B D
f
1; 4; 6
g
[
f
2; 4; 5
g
D
f
1; 2; 4; 5; 6
g
D
f
3
g
A B D
f
2; 3; 5
g
f
1; 3; 6
g
D
f
3
g
:
erefore, A [ B D A B D
f
3
g
.
According to the definition of the intersection of two sets, the complement of a set and
union of two sets, we have:
A B D
f
1; 4; 6
g
f
2; 4; 5
g
D
f
4
g
D
f
1; 2; 3; 5; 6
g
A [B D
f
2; 3; 5
g
[
f
1; 3; 6
g
D
f
1; 2; 3; 5; 6
g
:
erefore, A B D A [ B D
f
1; 2; 3; 5; 6
g
.
2.4 DEFINITION OF PROBABILITY
Probability is the measure of a likelihood that an event will occur. e calculation of probability
can be based on two typical definitions: the relative frequency and the axiomatic definition.
2.4.1 RELATIVE FREQUENCY
One classical approach to define a probability is the relative frequency. e probability of the
occurrence of an event E denoted as P .E/ is defined as the ratio of the number of occurrences
2.4. DEFINITION OF PROBABILITY 29
of the event A to the total number of trials.
P
.
E
/
D lim
N !1
n
N
; (2.9)
where n is the number of occurrences of the event E and N is the total number of trials of the
experiment. Since minimum and maximum possible values of n are 0 and N , we have:
0 P
.
E
/
1: (2.10)
is definition gives us a tool to calculate the probability of an event if a set of data of trials are
given.
Example 2.5
One person has tossed a coin 1,000 times, got 482 “heads” and 518 “tails.” Calculate the prob-
ability of showing “head” and the probability of showing “tail.”
Solution:
Let A represent the event of showing “head” and B the event of showing “tail.” So,
A D
f
head
g
; B D
f
tail
g
:
According to the relative frequency definition of probability, we have:
P .A/ D
482
100
D 0:482; P.B/ D
518
1000
D 0:518:
Example 2.6
A company designs and distributes a machine unit, which has a specified service life of two
years. e technical data shows that a total of 25,192 units have been sold and 1,053 units failed
during the two-year service. Estimate the probability of the failed unit and the safe unit during
the two-year service.
Solution:
Let A represent the event of failure during two-year service and B the event of working safely
during two-year service. So,
A D
f
Failure
g
; B D
f
Safe
g
:
According to the relative frequency definition of probability, the probability of the failed ma-
chine unit will be:
P
.
A
/
D P
.
failure
/
D
1053
25;192
D 0:042 D 4:2%:
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