Environmental Engineering Reference
In-Depth Information
Unlimited increase of
n
is associated with statistically ordering when the
variation of
the frequency of event
A
(
sample estimate
) becomes less and
less marked and approaches a constant value -
the probability of event
A
.
Basic rules of probability theory. Probability addition theorem.
If
A
1
,
A
2
, ...,
A
n
are incompatible events, and
A
is the sum of these events,
the probability of event
A
equals the sum of the probabilities of events
A
1
,
A
2
, ...,
A
n
:
{ }
n
n
∑
[1.6]
PA P A
(
)
=
PA
(
=
).
i
i
i
=
1
i
=
1
Because the opposite events
A
and
Ā
are incompatible and form a complete
group, the sum of their probabilities
PA PA
( ) +
(
) =1.
[1.7]
Probability multiplication theorem.
The probability of the product of
two events
A
1
and
A
2
is the probability of one of them, multiplied by the
conditional probability of another, assuming that the first event occurred:
PA A PAPA A PAPA A
(
∧
) = (
)
(
|
) = (
)
(
|
),
[1.8]
1
2
1
2
1
1
1
2
where the conditional probability of an event
A
1
at the beginning of event
A
2
is
the
probability of event
A
1
, calculated on the assumption that the event
occurred
A
2
:
PA A PA A P
⋅
[1.9]
For any finite number of events multiplication theorem takes the form
(
|
) = (
)
(
).
1 2
12
2
{ }
1
n
P A PA A A PA A A PA A PA
−
=
(
|
...
)
(
|
...
)...
(
|
)
(
).
[1.10]
i
2
n
2
3
n
n
1
n
n
i
=
1
If the events A
1
and A
2
are independent, then the corresponding conditional
probability is
PA A PA PA A PA
(
|
) = (
);
(
|
) =
(
),
1
2
1
2
1
2
so the multiplication theorem of probability [1.8] takes the form
PA A PAPA
(
∧
) =
(
)
(
),
[1.11]
1
2
1
2
and for a finite number of n independent events
{ }
{ }
n
n
=
∏
P A PA
=
.
[1.12]
i
i
i
1
i
=
1
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