42 2. FUNDAMENTAL RELIABILITY MATHEMATICS
en, the probability of a student with a passing grade who attended the tutoring section, that
is, P .T
j
A/, will be:
P .T
j
A/ D
P .A
j
T / P .T /
P .A/
D
0:95 0:18
0:73
D 0:234 D 23:4%:
Example 2.24
A product in a company is produced on three assembly lines. ree assembly lines account for
15%, 35%, and 50% of the product. e percentage of defective product items produced is 5%
for the first assembly line; 4% for the second assembly line; and 1% for the third assembly line.
If a product item is chosen at random from the total product output and is found to be defective,
what is the probability that it was produced by the third assembly line?
Solution:
Let events B
1
; B
2
, and B
3
to represent the first, second, and third assembly line. Event A rep-
resents the defective unit. According to the provided information, we have:
P
.
B
1
/
D 0:15I P .A
j
B
1
/ D 0:05I P
.
B
2
/
D 0:35I P .A
j
B
2
/ D 0:04I
P
.
B
3
/
D 0:50I P
.
A
j
B
3
/
D 0:01:
ese are prior probability based on information in the past. Now, we have a defective unit,
the probability of this defective unit, which is caused by the third assembly line is a posterior
probability and can be calculated as:
P
.
B
3
j
A
/
D
P
.
A
j
B
3
/
P
.
B
3
/
P
iD3
iD1
P
.
B
i
/
P
.
A
j
B
i
/
D
P
.
A
j
B
3
/
P
.
B
3
/
P
.
A
j
B
1
/
P
.
B
1
/
C P
.
A
j
B
2
/
P
.
B
2
/
C P
.
A
j
B
3
/
P
.
B
3
/
D
0:01 0:50
0:05 0:15 C 0:05 0:35 C0:01 0:50
D 0:189 D 18:9%:
2.6 RANDOM VARIABLE
e sample point (outcome) of an experiment can be described by a statement such as “head”
or “tail” in tossing a coin experiment or can be described by a numerical value such as showing
the number in rolling a dice experiment.
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