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I am not comfortable with the p of a binomial distribution.
Since the mean=p*n (using n for the number of attempts
and r for the number of successes) then p depends on n.
Since the mean=p*n (using n for the number of attempts
and r for the number of successes) then p depends on n.
This conflict with a view that p is an intrinsic property of
a setup.
Is it not true that p-->r/n when n-->oo? Defined
in that way it should not depend on n.
Example: trying to hit a target with an arrow. My skill
for being successful can be described by the probability
of a hit in a single trial. If p=0.2 then I am expected to
hit the target 2 out of 10 attempts. In reality the outcome
will fluctuate around 2. In this case n is given and all is
fine.
But in the case of a Geiger counter (where the mean was
2.20) both n=170 and n=340 give a satisfactory fit to
experimental data. This implies that both p=0.0129 and
p=0.00647 are OK. How can it be?
Isn't the objective
probability of recording one event per unit time equal
to 0.24 (7278/30058)?
Is the word "probability" used
for two different things in this "paradox"?