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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:
   if:  
   if:  
   if:  
   if:  

Without specifying the number of fractions, the worst case probability pmax becomes:

(2)

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  1. 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.
    p: %, f: , k:
  2. Calculating fact for a given p and k:

    Example: if k = 60 grains have been dated, the fraction that we know with (100-p) = 95% certainty not to have missed, is fact = 0.086
    k: , p: %, fact:
  3. Calculating pmax 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 pmax= 64%.
    k: , f: , pmax %
  4. Calculating Mopt 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 Mopt = 9 bins in the age-histogram.
    p: %, f: , k: , Mopt: