PHALIN THAKKAR
Twenty five percent, under which assumptions?
Which trials enter the denominator, and what does a finite sample show?
Note 6 · Explanatory note
Analytical result · Hypothetical examples are not physical measurements.
For the existing ideal baseline, half the sender and receiver bases match on average. Matching bases give the same bit. Mismatched bases give a random bit, with disagreement probability ½.
Retaining only matching bases changes the population: its ideal error rate is zero, while its expected size is half of all transmissions.
For N = 400 independent trials, the retained count has mean 200 and standard deviation 10. The all trial disagreement count has mean 100 and standard deviation √75 ≈ 8.66. Its fraction therefore has standard deviation about 0.0217, or 2.17 percentage points.
Those are distribution properties, not guaranteed counts from one run. The retained and disagreement counts should not be treated as independent quantities.
Extension: observing no errors in a tested sample does not establish that an unknown error probability is zero. For 20 independent tested bits and a true error probability of 0.10, the probability of seeing no errors is 0.9²⁰ ≈ 0.1216.