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intersection of the two events. It is also called multiplication formula.
(4) Total Probability Formula. Assume all the influence factors of event
A
are
B
1
,
B
2
,
…
, and they satisfy
B
i
·B
j
= ∅, (
i
¬
j
) and
P
(
∪
B
i
) = 1,
P
(
B
i
) > 0,
i
= 1, 2,
…
,
then we have:
B
i
) (6.4)
(5) Bayesian Formula. Bayesian formula, which is also called posterior
probability formula or inverse probability formula, has wide application.
Assume
P
(
A
) =
¶
P
(
B
i
)
P
(
A
|
P
(
B
i
) is prior probability, and
P
(
A
j
|
B
i
) is new information gained
from investigation, where
i
=1, 2,
…
,
n
, and
j
=1, 2,
…
,
m
. Then the posterior
probability calculated with Bayesian Formula is:
P B
(
)
P A
(
|
B
)
i
j
i
=
Ã
(6.5)
Example 6.3
One kind of product is made in a factory. Three work teams (
P B
(
|
A
)
i
j
m
P B
(
)
P A
(
|
B
)
k
j
k
k
=
1
A
1
,
A
2
,
and
A
3
) are in charge of two specifications (
B
1
and
B
2
) of the product. Their daily
outputs are listed in Table 6.2
Table 6.2 Daily outputs of three teams
B1
B2
Total
Team A1
2 000
1 000
3 000
Team A2
1 500
500
2 000
Team A3
500
500
1 000
Total
4 000
2 000
6 000
Now we randomly pick out one from the 6000 products. Please answer the
following questions.
1. Calculate the following probabilities with Classical Probability
(1) Calculate the probabilities that the picked product comes from the outputs of
A
1
, A
2
or
3
respectively.
Solution
P
A
(
A
1
)=3000/6000=1/2
P
(
A
2
)=2000/6000=1/3
3
)=1000/6000=1/6
Calculate the probabilities that the picked product belongs to
P
(
A
B
1
or
B
2
respectively
Solution
P
(
B
1
)=4000/6000=2/3
2
)=2000/6000=1/3
(2) Calculate the probability that the pick product is
P
(
B
B
1
and come from
A
1
