Geoscience Reference
In-Depth Information
Fig. 13.4 The probability
P
that
an annual maximum
flow with a known
probability
p
=
F
(
x
)
=
0
.
98, will
be exceeded after
exactly
k
years. This is
a 50 y event and the
probability that it will
be exceeded after 50 y
is 0.007 43. (See
Example 13.2.)
0.02
P
0.01
0
0
20
40
60
801
100
k
Fig. 13.5 The probability
P
that an annual
maximum flow with
a known probability
p
=
F
(
x
)
=
0
.
98,
will be exceeded
after
k
or fewer
years. This is a 50 y
event and the
probability that it
will be exceeded
before 50 y have
passed is 0.636. (See
Equation (13.30) and
Example 13.2.)
1
0.8
P
0.6
0.4
0.2
0
0
20
40
60
80
100
k
In general, a success can refer to a day without rain, a day with rain, or some other
desirable event, whatever the case may be. For the sake of illustration, let
p
denote
the probability that a day without rain be followed by a day without rain. In this case,
the form of Equation (13.29) arises again as the simple product of the probabilities
of a dry day being succeeded by a dry day for a sequence of
k
days, multiplied by
the probability of a dry
k
th day being succeeded by a rainy (
k
1)th day; in other
words, (13.29) represents the probability of experiencing a sequence of exactly
k
dry
days.
This product in (13.29) involves the assumption that the events are independent, or
which is the same, that
p
remains constant and does not change with
k
, the duration of
the period of dry days. Whether or not this is the case can be examined on the basis of
the following considerations. In the present context, the probability that the dry period
will last at most
k
days is the sum of the probabilities of all dry periods shorter than
+
