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        -----Original Message-----
        From:   Philidor11@aol.com [SMTP:Philidor11@aol.com]
        Sent:   Friday, November 06, 1998 3:25 PM
        To:     jjackson@interalpha.co.uk
        Cc:     dinosaur@usc.edu
        Subject:        Re: EARLY EVOLUTION OF 'BIRDS'

        In a message dated 11/6/98 12:19:53 PM Eastern Standard Time,
        jjackson@interalpha.co.uk writes:


        I'm following your argument slowly.  The Law of Averages, or
regression to the
        mean, says that if you start flipping a coin and heads come out 9
times in a
        row, and then you keep flipping thru the eons, sometime before the
sun burns
        out or blows up or both you will probably have 9 more tails than
heads.  But
        don't count on it.  The idea that 9 heads in a row affects the next
flip is
        called 'Gambler's Fallacy' and is a major money maker.
        The kind of meta-analysis you're talking about was recently used by
Shafer et
        al at Harvard to look at 20 years, give or take, of pathological
        prevalence studies.  He observes the inadequacies of each approach
used, then
        tries to combine them to draw a conclusion that pathological
gambling has
        increased in recent years.  His reasoning is that because some
        studies before a certain date he picked produced some results just
outside the
        high end of the range of the inadequate studies after that date,
        pathological gambling has increased.  The difference in ranges is in
        hundredths of a percent.
        More relevantly, your mickles had better had some validity to begin
with.  I
        don't think inadequate results gain strength in numbers.  Validity
in this
        case refers to the likelihood that a fossil will be discovered.
        assuming both that each of the creatures you're comparing were
equally likely
        to fossilize and that the places where they were discovered were a
        representative sample of the large area you're concerned with. 
        Hi, All;

           Here's the 'short n' sweet' of the ole coin toss:
        Each time you toss the coin (IF there are no limiting factors, such
as cheating somehow) there is a 50/50 chance that it will land heads up or
tails up.  Previous tosses & future tosses have no bearing on the outcome.
There is a mathematical tendency to approach a mean state.  That is, as
Brian states; if you toss your coin over a long enough period of time, the
rate of occurrence will be APPROXIMATELY 50%.  BUT - how does one define the
length of time needed to approach the mean state.  With a coin toss, about
20 tosses should do it.  All things being equal, 9 heads or tails in a row
CAN occur, but will be rare.  Sampling is the issue & with fossilization it
seems that sampling is skewed automatically.  For example: dinosaur fossils
preserved from what were mountainous environments are rare.  This doesn't
mean that some dinosaurs weren't as adept at mountainous living as modern
day goats.  It MIGHT mean that, but it might equally mean that the
conditions for preservation are less conducive from mountainous regions.
This might be a poor example, but the point is that statistics can be very
skewed or even rendered inaccurate by sampling.  This is complicated by
interstitual or matrix inbedded chaos.  No matter how ordered a system is,
there is always an element of chaos inbedded within the structure.  The
reverse is also true.  In a complex system, which all biological sytems are,
there are numerous
        Elements of chaos (or unpredictability) within the structure.
        One element would be variation in animal behavior.  The point being
that a sampling population could seen intuitively robust, but in fact be
seriously skewed to produce one outcome or another.