4. Two proofreaders A and B, after reading the proof, found $a$ and $b$ typos, respectively, of which $c$ turned out to be common. Assuming that the proofreaders worked independently of each other, give a "reasonable" estimate of the number of remaining undetected typos.
Solution. Consider the following statistical model. $2 n$ independent Bernoulli trials $\xi_{1}, \ldots, \xi_{2 n}$ are conducted. The event $\left\{\xi_{2 k}=1\right\}$ means that proofreader $A$ noticed the $k$-th typo, $\left\{\xi_{2 k-1}=1\right\}$ means that proofreader $B$ noticed the $k$-th typo, $k=1, \ldots, n$.

Since $n$ is large, we can assume that $p_{a}=\mathrm{P}\left\{\xi_{1}=1\right\}=a / n$ and $p_{b}=\mathrm{P}\left\{\xi_{2}=1\right\}=b / n$.

Due to independence, the probability that both proofreaders notice the same typo is $p_{a} p_{b}=\frac{a b}{n^{2}}$. On the other hand, the proportion of typos noticed by both proofreaders is $c / n$, so for large $n$ we can assume that $\frac{a b}{n^{2}}=\frac{c}{n}$, i.e., $n=a b / c$.

By the inclusion-exclusion principle, the number of typos that remain undetected is

$$
n-a-b+c=\frac{a b}{c}-a-b+c .
$$

## § 8. Conditional Probabilities and Mathematical Expectations Relative to Partitions