I’ve figured out what your formula refers to that you never explicitly explained. Stated in English, your error rate is derived from the expected number of false positives.
If there is a 1 in 100,000 chance of selecting a black token and the reporting error rate is 1 in 1,000 then in 100,000,000 iterations the total number of errors will be 100,000,000/1,000 or 100,000 errors.
Since the ratio of black tokens to white ones is 100,000 to 1, 99,999 white tokens will be erroneously reported as black and 1 black token will be erroneously reported as white.
Further, since the ratio of black tokens to white tokens is 100,000 to one, out of 100,000,000 draws 1,000 black tokens will be accurately reported as black.
99,999 false black reports + 1,000 valid black reports = 109,999 total black token reports of which only 1,000 will be valid black tokens or a ratio of 1,000 valid black reports/109,999 total black reports or approximately 9.1% of the black reports will represent actual black draws and 90.9% of the black reports will represent white tokens erroneously reported as black.