2.5. SOME BASIC OPERATIONS OF PROBABILITY 37
the first picking and the blue ball in the second picking is
P
.
E
1
E
2
/
D P
.
E
1
/
P
.
E
2
/
D
3
7
4
7
D
12
49
:
Example 2.17
A water pump is driven by an electric motor. e reliability of the water pump is 0.95. e
reliability of the electrical motor is 0.99. Calculate the reliability of this unit, including an electric
motor and water pump.
Solution:
Let event E
1
to represent that the electric motor works properly and event E
2
that the water
pump properly works. According to the given information, we have:
P
.
E
1
/
D 0:99; P
.
E
2
/
D 0:95:
e electrical motor and the water pump are two different products. ey function according to
their working principles. So, it is obvious that E
1
and E
2
are statistically independent. erefore,
the probability of the unit, including the electric motor and the water pump is:
P
.
E
1
E
2
/
D P
.
E
1
/
P
.
E
2
/
D 0:99 0:95 D 0:94:
2.5.6 CONDITIONAL PROBABILITY
e occurrence of the event E
2
when the event E
1
has occurred is known as the conditional
event and is denoted as E
2
j
E
1
. e conditional probability of this conditional event E
2
j
E
1
is
defined as
P
.
E
2
j
E
1
/
D
P
.
E
2
E
1
/
P
.
E
1
/
P
.
E
1
/
> 0: (2.21)
Equation (2.21) can also be expressed as
P
.
E
2
E
1
/
D P
.
E
2
j
E
1
/
P
.
E
1
/
D P
.
E
1
j
E
2
/
P
.
E
2
/
: (2.22)
If these two events E
1
and E
2
are statistically independent, we can have
P
.
E
2
j
E
1
/
D P
.
E
2
/
: (2.23)
Example 2.18
An engineering class has 67 students. Twenty-seven students choose a mechanical-concentrated
program, 16 students choose an electrical-concentrated program, and the remaining 24 students
38 2. FUNDAMENTAL RELIABILITY MATHEMATICS
Table 2.1: e data for enrollments of two classes
Event M Event E Event G
Total 27 16 24
Event A 10 188
Event B 17 8 6
a general engineering program. e data about the registration of a science class, an engineer-
ing class, and corresponding event symbols are shown in Table 2.1. Event M is mechanical-
concentrated; event E is electrical-concentrated; event G is general engineering. Event A is a
science class, and event B is an engineering class. Calculate P .M A/
Solution:
According to the provided information, we have:
P
.
M
/
D
27
67
; P
.
A
/
D
10 C8 C 18
67
D
36
67
:
According to the definition of the conditional probability and the provided information, we
have:
P
.
A
j
M
/
D
10
27
; P
.
M
j
A
/
D
10
10 C8 C 18
D
10
36
:
From Equation (2.22), we have
P
.
M A
/
D P
.
A
j
M
/
P
.
M
/
D
10
27
27
67
D
10
67
P
.
M A
/
D P
.
M
j
A
/
P
.
A
/
D
10
36
36
67
D
10
67
:
Example 2.19
In a company, a well-planned design project is denoted as event A, and a well-executed project is
denoted as event B. If the probability that a design project will be well planned is 0.75, and the
probability that the design project will be well planned and well executed is 0.70. Calculate the
probability of a well-planned design project that will also be well-executed, that is, P
.
B
j
A
/
.
Solution:
According to the given information, we have:
P
.
A
/
D 0:75; P
.
B A
/
D 0:70:
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