How many grains are needed for a provenance study?
The general equation for the calculation of the
probability p that at least one of M fractions >= f of the population
is missed by all k grains of a sample is:

(1) 
with:
Without specifying the number of fractions, the worst case probability p_{max} becomes:

(2) 
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Calculating k as a function of the desired p and f:
Example: to reduce the chance that at least one fraction f >= 0.05 has been missed
to less than p = 5%, k = 117 grains have to be dated.
Calculating f_{act} for a given p and k:
Example: if k = 60 grains have been dated, the fraction that
we know with (100p) = 95% certainty not to have missed,
is f_{act} = 0.086
Calculating p_{max} as a function of k and f:
Example: when k = 60 grains are dated of a uniformly distributed population, the chance that at least one
fraction f >= 0.05 has been missed is p_{max}= 64%.
Calculating M_{opt} for a given p, f and k:
Example: if we want to reduce the chance that at least one fraction f >= 0.05 was missed
of a uniform population to less than p = 5%, and we have dated k = 100 grains, then
we can only use M_{opt} = 9 bins in the agehistogram.
