Proof that the derivative of the prime counting function is the probability of prime? The derivative of the estimation of the prime counting function, $\frac{x}{ln(x)}$, is $\frac{ln(x)-1}{ln(x)^2}$, which is approximately $\frac{1}{lnx}$ for large values of $x$. According to the prime number theory, $\frac{1}{lnx}$ is the probability that a randomly chosen integer between 2 and $x$ is a prime number.
Why is the derivative of the prime counting function the probability of getting a prime number?
Edit: I've only studied up to basic integrals, so try to keep it simple!

You are describing Cramer's model of the primes, which is pretty good. However, Maier showed about 1985 that it gave incorrect estimates for short intervals. See if i can find it, it's a famous episode. 
Maier's theorem
