Dividing by 3

It's intuitive that the probability that a random integer is divisible $p$ should be about $\frac{1}{p}.$

However, to get a true probability and avoid issues with infinite sets, we are looking at $\left[1,99^{99}\right].$ Every 3rd number will be divisible by 3 ( $3,6,9,\ldots,99^{99}$ ), so the probability that a randomly chosen integer in this interval is divisibly by 3 is $\frac{1}{3}.$

In general, if the probability that a random integer in $[1,n]$ is divisible by $p$ is $f(n,p)$ , then $\lim_{n\rightarrow\infty} f(n,p) = \frac{1}{p}.$ In other words, the limiting probability is $\frac{1}{p}.$