5. Arithmetic density. It is clear that if we take a random natural number from a sufficiently long segment of the natural number series, the probability that this number will be divisible by 10 will be closer to 0.1, the longer the segment.

Let's ask a different question: how much can this probability differ from 0.1? It can be 0, for example, if we specified the segment $1,2,3$, where no number is divisible by 10. And what is:

a) (recommended for 6th grade, 1 point) the greatest possible probability;

b) (recommended for 6th grade, 1 point) the smallest possible non-zero probability?
Solution. a) The greatest probability is 1. For this, it is sufficient to take the single number 10 as the segment.

b) Let there be $k \geq 1$ numbers divisible by 10 in the chosen segment. The greatest length of such a segment is $10 k+9$. Therefore, the smallest probability of the event "the chosen number is divisible by 10" is

$$
p=\frac{k}{10 k+9}=\frac{1}{10}-\frac{9}{10(10 k+9)}.
$$

The smallest value is achieved for the smallest $k$, i.e., for $k=1$. We get: $p=\frac{1}{19}$.

Answer: $1 / 19$.