Re: a simple(?) probability question...


James Waldby wrote:
Joe wrote:
[...] By definition, a 100 Years Storm is a storm that will occur at
least once in 100 years. That means certainty within 100 years, i.e.
probability equals 1 in the elapsed time of 100 years.
...

That doesn't agree with common terminology as stated, for example,
in http://bcn.boulder.co.us/basin/watershed/flood.html and
http://en.wikipedia.org/wiki/100-year_flood . From the former:
The terms "10 year", "50 year", "100 year" and "500 year" floods
are used to describe the estimated probability of a flood event
happening in any given year. [...] A 10 year flood has
a 10 percent probability of occurring in any given year, a 50
year event a 2% probabaility, a 100 year event a 1% probability,
and a 500 year event a .2% probability. While unlikely, it is
possible to have two 100 or even 500 year floods within years or
months of each other.

Given the likelihood of 1% for a 100-year event to occur in a
given year, one could figure 1 - (1-0.01)^50 or ~ .395 likelihood
for it to occur at least once in any given 50-year period, or
1 - (1-0.01)^100 or ~ .634 likelihood for any 100-year period.

A Poisson process (http://en.wikipedia.org/wiki/Poisson_process )
with lambda = 0.01 per year has probability ~ .632 of 1 or more
events occurring in any 100-year period, which is about the same
as for the Bernoulli process mentioned in the preceding paragraph,
just as one would expect for a small probability and many years.

-jiw

I think the debated definitions have the same consequence from a
probability perspective. A ten year storm that has a probability of
10% in one year adds to a 100% in 10 years, i.e. probability of 1.0
(certainty). Likewise the 100 year storm that has a probability of 1%
in one year adds to 100% in 100 years. It also adds to 50%, i.e.
probability of .5 at the half way point.

On the question of continuous and discrete, I think there is a clear
distinction to be made between a repetition of events in a time period,
i.e. say like a line of people approaching a service center as opposed
to a single discrete event that occurs within a certain interval of
time say like an eclipse of the sun, for example. It is true that in
both cases the event either exists or does not exist at a certain time,
however in the former the process in place ensures that there will be a
stream of events, i.e.it is a continuous process, whereas in the latter
case only one event will occur, i.e. the discrete event. The Poisson
distribution has general application to such continuos processes, it is
not very well suited to estimating probabilities of discrete rare
events. The Bernoulli model is better for this situation. see E.R.
Dougherty "Probability and Statistics for the Engineering, Computing,
and Physical Sciences" Prentice Hall, 1990 pp152

After all is said and done, the concept of 100 year storm and its
liklihood is a very tenuous one. Likewise, probability theory itself
has certain weakness. There is no such thing as a probability of 1.0,
i.e. nothing is absolutely certain and the same applies to zero
probability since nothing is absolutely completely uncertain. But we
use these statistics all the time. Probability theory just defines a
mathematical construct for estimating the likehood of something, given
a certain data set used as an estimate for the universe of events of
interest. The 100 year storm, by its very nature is an event for which
there cannot be very much data. It is a rare event, and the number of
100 year cycles for which there is recorded climate data is small. So
whoever it was that made the specification of what contitutes a 100
year storm must have used a lot of subjective judgement. If that
judgement included the idea that a storm of a certain high intensity
could happen in any year over a certain number of years, then that
judgement described a process having a normal (Gaussian) distribution.

.

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