Information Technology Reference
In-Depth Information
proportional to the volume of the domain, and is independent of the position or
the shape of the domain in
. Under the restrictions above, we call a trial a
geometrical random trial, where
is basic event space.
Event in geometrical random trial: Assume that
is the basic event space of
geometrical random trial, and A is a subset of
that can be measured with
volume, where L(A) is the M-dimensional volume of A. Then the event of
“random point falls in domain A” is represented with A. In
, a subset that can
be measured with volume is called a measurable set. Each measurable set can be
viewed as an event. The set of all measurable subsets is represented by F.
Definition 6.3
Geometrical Probability: Assume
Ȫ
is a basic event space of a
geometrical random trial, and F is the set of all measurable subsets of
Ȫ
. Then
the probability of any event A in F is the ratio between the volume of A and that
of
Ȫ
.
P(A) = V(A)/V(
)
(6.3)
Definition 6.4
Conditional Probability: The probability of event A under the
condition that event B has happened is denoted by P(A|B). We call it the
conditional probability of event A under condition B. P(A) is called
unconditional probability.
Example 6.2
There are two white balls and one black ball in a bag. Now we take
out two balls in turn. Questions: (1) How much is the probability of the event that
a white ball is picked in the first time? (2) How much is the probability of the
event that a white ball is picked in the second time when a white ball has been
picked in the first time?
Solution:
Assume
A
is the event that a white ball is picked in the first time, and
B
} is the
event that a white ball is picked in the second time when a white ball has been
picked in the first time. According to Definition 6.4 we have:
is the event that a white ball is picked in the second time. Then {
B|A
(1) No matter under repeated sampling or non-repeated sampling,
P
(
A
)=2/3
(2) When sampling is non-repeated,
P
(
B|A
)=1/2; When sampling is repeated,
P
(
B|A
) =
P
(
B
) = 2/3. The conditional probability equals to non-conditional
probability.
If the appearance of any of event
A
or
B
will not affect the probability of the
other event, viz.
P
(
A
) =
P
(
A|B
) or
P
(
B
) =
P
(
B|A
). We call event
A
and
B
independent events.
