One can use the results of two independent quality tests to improve the estimate of the probability of a finding a good item. The counts for Tester 1 of good and bad items are sesignated N₁ and N₂ and those of Tester 2 are N₃ and N₄ and their probabilities are p and q and p' and q' respectively. Based on the two observers' assessments of the items how can we determine NG, NB, pG and pB? The answer is to keep track of the number of times, N13, when both N1 and N3 are good and the number of times, N24, when both N2 and N4 are bad.
Then we can borrow a trick used by Rutherford to improve on the scintillation rates determined by two different observers.
Even with relatively large rates for the counting errors one still get a good estimate of the actual rates.
The mean values indicated were found by averaging the counts for NG and NB using NG+NB=N0.
Supplemental (Sep 26): The estimated probabilities for G and B are again the averages for the 10 sets of assessments each involving 100 items. The set of stochastic variables used to generate the data were for pG, a, d, a' and d' and were randomly set to 1 or 0 based on the rates as was done previously.
Supplemental (Sep 26): See Feller, An Introduction to Probability Theory and Its Applications, Vol I, 2nd ed., p. 160, prob. 23 which cites Rutherford. See also Rutherford &al., Probability Variations in the Distribution of α Particles cited in Rutherford's book linked above.
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