Correcting Faulty Observations

  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 N_G, N_B, p_G and p_B? 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 N_G and N_B using N_G+N_B=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 p_G, 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.