It is known that in the small lottery, 5 numbers are drawn from 90. If someone bets on 1, 2, 3, 4, or 5 numbers, what is the probability that these numbers will be drawn? In the past, when the small lottery was still in operation, the state paid out 14, 240, 4800, 60000, and 100000 times the stake for estratra (one number), ambora (two numbers), ternora, quaternora, and quinternora, respectively.

What percentage of all stakes was the state's revenue?
$$
\begin{gathered}
v_{1}=\binom{5}{1}:\binom{90}{1}=\frac{1}{18} \\
v_{2}=\binom{5}{2}:\binom{90}{2}=\frac{2}{801} \\
v_{3}=\binom{5}{3}:\binom{90}{3}=\frac{1}{11748} \\
v_{4}=\binom{5}{4}:\binom{90}{4}=\frac{1}{511038} \\
v_{5}=\binom{5}{5}:\binom{90}{5}=\frac{1}{43949268}
\end{gathered}
$$

Since the state paid out 14 times the deposit, the profit was

$$
\text { (1) } \frac{18-14}{18} \cdot 100 \%=\left(1-14 \times \frac{1}{18}\right) \cdot 100 \%=\left(1-14 v_{1}\right) 100 \%=22.22 \%
$$

Similarly

(2) $\quad\left(1-240 v_{2}\right) 100 \%=40.07 \%$

(3) $\left(1-4800 v_{3}\right) 100 \%=59.14 \%$

(4) $\quad\left(1-60000 v_{4}\right) 100 \%=88.26 \%$

(5) $\quad\left(1-100000 v_{5}\right) 100 \%=99.78 \%$

(Alfréd Haar, Budapest.)

The problem was also solved by: Dömény I., Krampera Gy., Pető L., Rosenberg J., Schöffer I., Sonnenfeld J.