Relative Frequency is Probability Measure

Theorem
The relative frequency model is a probability measure.

Proof
We check all the Kolmogorov axioms in turn:

First Axiom
Let $n$ be amount of times a certain event has been observed.

Let $n'$ be the amount of observations where the event could have happened, but did not.

By construction, $n$ and $n'$ are natural numbers, and provided that some observation was done, $n + n' \ne 0$.

The relative frequency model says that the probability of a event $E$ can be defined as


 * $\Pr \left({E}\right) = \dfrac n {n + n'}$

If the event was observed never to happen, $n = 0$, and then the probability would be defined as $\dfrac 0 {0 + n'} = 0$.

If the event was observed to happen every time, $n' = 0$, and then the probability would be defined as $\dfrac n {n + 0} = 1$.

Any other combination of observations would have $\dfrac n {n + n'}$ be a fraction, the numerator of which is a positive number, and the denominator of which is a higher positive number, because $n < n + n'$.

Thus the image of $\Pr$ is bounded as such:


 * $ 0 \le \Pr\left(\cdot\right) \le 1$

Second Axiom
By hypothesis,


 * $\Pr \left({\Omega}\right) = \dfrac {n + n'}{n + n'} = 1$.