Anne, Edward, and Isaac are the three editors of the Perplexus Weekly newspaper. This week, Anne spotted 300 errors, Edward spotted 100 errors, and Isaac spotted 200 errors. Altogether, they spotted 404 errors. Assuming that each error is equally easy to spot, about how many errors did they miss?

Assuming that the numbers observed are proportional to the probabilities that the individuals would spot errors, and that the error spotting is independent among the editors, and x is the number of errors altogether, the probability that an error is found (404/x) is the sum of the probabilities of each editor finding the error (100/x + 200/x + 300/x), minus the sum of the probabilities of pairwise finding (100*200/x^2 + 200*300/x^2 + 100*300/x^2), plus the probability that all would find the given error (100*200*300/x^3). So we need to solve:

404/x = 100/x + 200/x + 300/x - (100*200/x^2 + 200*300/x^2 + 100*300/x^2) + 100*200*300/x^3

The above equation is satisfied by x = 500. Since 404 errors were found, most likely 96 were not found. BTW, in this scenario 24 errors were found by all 3 editors. 40 were found by Edward and Isaac (including the 24 found by all); 60 were found by Anne and Edward and 120 by Anne and Isaac.

x = 555
FOR i = 1 TO 40
    x2 = x * x: x3 = x2 * x
    v = 100 / x + 200 / x + 300 / x - 20000 / x2 - 30000 / x2 - 60000 / x2 + 6000000 / x3
    x = 404 / v
    PRINT x
NEXT

which converges to 500:

 532.2342
 518.7455
 510.8466
 506.2567
 503.6024
 502.0718
 501.1908
 500.6841
 500.393
 500.2257
 500.1296
 500.0745
 500.0428
 500.0246
 500.0142
 500.0081
 500.0047
 500.0027
 500.0016
 500.0009
 500.0005
 500.0003
 500.0002
 500.0001
 500.0001
 500
 500
 500
 500

Posted by Charlie on 2006-02-14 14:44:47