Proofreading Mistakes

If proofreader $A$ finds $a$ mistakes and proofreader $B$ finds $b$ mistakes, $c$ of which were also found by $A$, how many mistakes were missed by both $A$ and $B$? Assume there are a total of $m$ mistakes, so proofreader $A$ finds a Fraction $a/m$ of all mistakes, and also a Fraction $c/b$ of the mistakes found by $B$. Assuming these fractions are the same, then solving for $m$ gives

$$m={ab\over c}.$$

The number of mistakes missed by both is therefore approximately

$$N=m-a-b+c = {(a-c)(b-c)\over c}.$$

References

Pólya, G. ``Probabilities in Proofreading.'' Amer. Math. Monthly, 83, 42, 1976.