Prime number theorem

The number of Primes $\pi (n)$ smaller than $n\in {\mathbb{Z}}^{+}$ satisfies $\pi (n)\approx \frac{n}{\log (n)}$.

Proof sketch

The probability that a number $p$ is Prime is approximately $\frac{1}{p}$. Assuming that each number being Prime forms a family of independent events, the result follows. This assumption is questionable, as there are no Random variables determining whether a number is prime, so more thorough proofs of this are needed. Nonetheless, this should provide an intuition on how this value was obtained.