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HYB: Rolling the Dice

From: Sharon McAllister <73372.1745@compuserve.com>

Gerry wrote:

>  I don't think it's quite that bad.

>  The probability of any individual seedling failing to meet the criteria
is .999,
>  so the probability of a thousand seedlings all failing is .999 raised to
>  1000th power, which seems to come out to .368 .

>  Therefore the chance of success is .632 if 1000 seedlings are grown.

>  I think the 50/50 point is reached at 693 seedlings.

So you're not going to let me get away with avoiding the math this time
<Sigh> ....

The problem arises if we assume that the "probability of any individual
seedling failing to meet the criteria is .999" -- because we simply don't
have the data to support it.   That's why I didn't approach the question in
terms of statistical probabilities.  All we have is a hybridizers' Rule of
Thumb -- we're dealing with anecdotal perceptions reported by
non-statisticians and, to the practicing hybridizer, the 1 in 1,000 figure
is an instinctive "average".  Actually one of many estimates, depending on
who is expressing an opinion.

Look at it this way -- if there's one "success" per thousand seedlings then
for every case in which successes are separated by less than 1000 there's a
matching case in which successes are separated by more than 1000.   If you
grow 1000 seedlings, you have a 50/50 chance of success.  If you grow 2000,
you can expect one good seedling and a 50/50 chance of a second one.   And
so on, per thousand seedlings.  Simple.  Intuitive.  And completely
ungrounded in statistics.

The statistical model, OTOH, isn't simple at all.   To those of you who are
mathematically inclined, I suggest setting up a spreadsheet for the model
and graphing the distribution.  Most enlightening.  In summary, it shows
that not only will some successes be separated by considerably less than
1000 and some by many more -- but also that the probability only
asymptotically approaches certainty.  The good news is that, as Gerry
demonstrated, the 50/50 point is skewed in our favor.  The bad news is that
in this model it would take around 5000 seedlings to reach the 99% chance
of success.  

So how can we reconcile the two?  

First, we note that hybridizers avoid exploring the outer limits of the
statistical model.  We learn to capitalize on success and cut our losses
when confronted with a string of failures.   When we are fortunate enough
to achieve early success, we don't feel compelled to raise the number of
seedlings needed to round out the lot. In short, we do everything we can to
influence the odds in our favor.  I believe these practices have led us to
an excessively optimistic estimate of the overall probability of success
for any one seedling.   If the statistical model used .9995 as the
probability of failure, the 50/50 point would be  nearer to 1000 seedlings
and I believe this would more accurately depict empirical observations.

But even that is a guesstimate.  We're rolling the dice.  The important
thing is to make the crosses, grow the seedlings and have fun!

Sharon McAllister

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