32 2. FUNDAMENTAL RELIABILITY MATHEMATICS
the corresponding calculations based on that all sample points have the exact occurrence
likelihood. In this example, we have a total of 7 possible sample points, and 3 of them are
the occurrence of a white ball. So, P
.
whiteball
/
D
3
7
. However, the probability by the rel-
ative frequency definition will depend on the total number of experiments. When the total
number of the experiment is toward an infinite, the probability calculated by the relative
frequency definition will be equal to that by the axiomatic definition.
Example 2.9
For the experiment of rolling two dice, calculate the probability of the sum of two numbers less
than 5.
Solution:
For the experiment of rolling 2 dice, there is a total of 36 possible sample points (outcomes).
e sum of two numbers less than 5 will only include following 6 sample points: f1; 3g, f3; 1g,
f2; 2g, f1; 2g, f2; 1g, and f1; 1g. erefore, the probability of the sum of two numbers less than
5 will be:
P
.
sum less than 5
/
D
6
36
D
1
6
:
2.5 SOME BASIC OPERATIONS OF PROBABILITY
Based on the axiomatic definition of probability and the set theory, we have the following simple
operations of the probability.
2.5.1 PROBABILITY OF MUTUALLY EXCLUSIVE EVENTS
When the two sets (events) E
1
and E
2
are mutually exclusive events, that is, no common el-
ements as shown in Figure 2.4, the probability P
.
E
1
E
2
/
of the intersection of mutually
exclusive events will be zero:
P
.
E
1
/
D P
.
;
/
D 0: (2.14)
e probability of two mutually exclusive events will be:
P
.
E
1
[ E
2
/
D P
.
E
1
/
C P
.
E
2
/
: (2.15)
E
1
E
2
Figure 2.4: e Venn diagram of two mutually exclusive events.
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