1685. The total amount of all deposits in a certain savings office is 20000 rubles, and the probability that a randomly selected deposit does not exceed 100 rubles is 0.8. What can be said about the number of depositors in this savings office?
Solution. Let $X$ be the amount of a randomly selected deposit, and $n-$ the number of all depositors. Then, from the condition of the problem, it follows that $E X=\frac{20000}{n}$. Since $P(X<100)=0.8$ and by Markov's inequality $P(X<100) \geqslant 1-\frac{E X}{100}$, then $0.8 \geqslant 1-\frac{20000}{n \cdot 100}$. From this,

$$
\frac{20000}{n \cdot 100} \geqslant 0.2, 200 \geqslant n \cdot 0.2, n \leqslant 1000
$$