Information Technology Reference
In-Depth Information
Theorem 6.1
(Addition Theorem) The probability of the sum of two mutually
exclusive events equals to the sum of the probabilities of the two events. That is
P(A+B) = P(A)
P(B)
The sum of probabilities of two mutually inverse events is 1. In another word,
if
−1
−
A
+
A
=
Ω
A
A
A
, and
and
are mutually inverse, then P(
)
−
−
A
A
A
P(
)
If A and B are two arbitrary events, then
P
)=1, or P(
)=1-P(
)
holds. This theorem can be generalized to the case that involves more than three
events.
(
A+B
)=
P
(
A
)
P
(
B
)
P
(
AB
P
(
A+B+C
)=
P
(
A
)+
P
(
B
)+
P
(
C
)-
P
(
AB
)-
P
(
BC
)-
P
(
CA
)+
P
(
ABC
)
(Multiplication Theorem) Assume A and B are two mutually
independent non-zero events, then the probability of the multiple event equals to
the multiplication of probabilities of event A and B, that is:
P(A·B)=P(A)·P(B) or P(A·B)=P(B)·P(A)
Theorem 6.2
Assume
are two arbitrary non-zero events, then the probability of the
multiple event equals to the multiplication of the probability of event
A
and
B
A
(or
B
)
and the conditional probability of event
B
(or
A
) under condition
A
(or
B
).
P
(
A·B
)=P(
A
)·P(
B
|
A
) or
P
(
A·B
)=
P
(
B
)·
P
(
A
|
B
)
This theorem can be generalized to the case that involves more than three
events. When the probability of multiple event
P
(
A
1
A
2
…A
n
-1
)0, we have:
P
(
A
1
A
2
…A
n
)=
P
(
A
1
)·
P
(
A
2
|
A
1
)·
P
(
A
3
|
A
A
2
) …
P
(
A
n
|
A
A
A
n
-1
)
1
1
2
If all the events are pairwise independent, we have:
P
(
A
1
A
2
…A
n
)=
P
(
A
1
)·
P
(
A
2
)·
P
(
A
3
) …
P
(
A
n
)
6.2.2
Bayesian probability
(1) Prior Probability. A prior probability is the probability of an event that is
gained from historical materials or subjective judgments. It is not verified and is
estimated in the absence of evidence. So it is called prior probability. There are
two kinds of prior probabilities. One is objective prior probability, which is
calculated according to historical materials; the other is subjective prior
probability, which is estimated purely based on the subjective experience when
historical material is absent or incomplete.
(2) Posterior Probability. A posterior probability is the probability that is
computed according to the prior probability and additional information from
investigation via Bayesian Formula.
(3) Joint Probability. The joint probability of two events is the probability of the
